Inclusion Criteria:

          -  Any histologically confirmed recurrent endometrial adenocarcinoma (except for
             carcinosarcoma), recurrent high-grade serous ovarian/primary peritoneal/fallopian tube
             carcinoma, or deleterious BRCA mutant recurrent ovarian/primary peritoneal/fallopian
             tube cancer for whom no curative option is available will be eligible. DOSE ESCALATION
             ONLY: For patients with ovarian cancer in the dose escalation cohort, only patients
             with recurrent, platinum resistant disease are eligible to enroll on study. Platinum
             resistance is defined as progression within 6 months from completion of platinum based
             therapy (the date should be calculated from the last administered dose of platinum
             therapy). In addition, patients in this cohort with a BRCA mutation must have also
             progressed after treatment with PARP inhibitor.

          -  Patients may have unlimited prior chemotherapeutic regimens for management of
             recurrent endometrial or ovarian carcinoma. Patients who have received prior PARP
             inhibitors ARE allowed to participate. Patients who have received prior PI3K-pathway
             inhibitors ARE allowed to participate on the dose escalation phase ONLY. Patients may
             have progressed on prior PARP inhibitor and/or PI3K-pathway inhibitor but they may not
             have discontinued drug for toxicity.

          -  With the exception of alopecia, peripheral neuropathy, and bone marrow parameters, any
             unresolved toxicities from prior chemotherapy should be no greater than Common
             Terminology Criteria for Adverse Events (CTCAE) (version 5.0) grade 1 at the time of
             starting study treatment.

          -  Patients should have measurable disease as defined by Response Evaluation Criteria in
             Solid Tumors (RECIST) 1.1. If no measurable disease is present, patients should have
             assessable disease such as pleural effusion, ascites, with CA125 Gynecological Cancer
             Intergroup (GCIG) criteria.

          -  Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of
             0 or 1.

          -  The effects of niraparib and copanlisib on the developing human fetus are unknown. For
             this reason, women of childbearing potential must have a negative urine or serum
             pregnancy test within 7 days prior to taking study treatment. Female patients and
             their male partners must also agree to use effective contraception when sexually
             active. This applies since signing of the informed consent form and 6 months (for
             women of child bearing potential) after the last study drug administration. A woman is
             considered of childbearing potential (WOCBP), i.e. fertile, following menarche and
             until becoming post-menopausal unless permanently sterile. Permanent sterilization
             methods include but are not limited to hysterectomy, bilateral salpingectomy and
             bilateral oophorectomy. A postmenopausal state is defined as no menses for 12 months
             without an alternative medical cause. A high follicle stimulating hormone (FSH) level
             in the postmenopausal range may be used to confirm a post-menopausal state in women
             not using hormonal contraception or hormonal replacement therapy.

          -  A man is considered fertile after puberty unless permanently sterile by bilateral
             orchidectomy. The investigator or a designated associate must advise the patient
             (WOCBP or men who have not undergone bilateral orchidectomy) how to achieve highly
             effective birth control method. Birth control should be used from the signing of the
             patient consent form and for 180 days following the last dose of niraparib. Acceptable
             methods of birth control include:

               -  Two highly effective forms of contraception, defined as contraceptive methods
                  with a failure rate of less than 1% per year when used consistently and
                  correctly. Patients and their sexual partners who've undergone vasectomy or tubal
                  occlusion must also use a male condom with spermicide.

               -  Permanent sterilization, defined as hysterectomy, bilateral salpingectomy,
                  bilateral oophorectomy, or bilateral orchidectomy; postmenopausal, defined as a
                  female patient or sexual partner > 45 years of age who has not menstruated for at
                  least 12 consecutive months; total sexual abstinence.

          -  It is unknown if niraparib or copanlisib are expressed in human breast milk. For this
             reason, women must not breast-feed while taking the study medications.

          -  Absolute neutrophil count >= 1,500/mcL (within 7 days prior to entry/randomization).

          -  Hemoglobin >= 9 gm/dL (within 7 days prior to entry/randomization).

          -  Platelets >= 100,000/mcL (within 7 days prior to entry/randomization).

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (within 7 days
             prior to entry/randomization) (< 2 x ULN for patients with Gilbert-Meulengracht
             syndrome or for patients with cholestasis due to compressive adenopathies of the
             hepatic hilum).

          -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 2.5 x ULN unless
             the liver is involved with tumor, in that case, ALT/AST must be =< 5 x ULN (within 7
             days prior to entry/randomization).

          -  Lipase =< 1.5 x ULN (within 7 days prior to entry/randomization).

          -  International normalized ratio (INR) =< 1.5 x ULN and partial thromboplastin time
             (PTT) =< 1.5 x ULN (within 7 days prior to entry/randomization). Prothrombin time (PT)
             can be used instead of INR if PTT =< 1.5 x ULN.

          -  Left ventricular ejection fraction (LVEF) >= 50% (within 7 days prior to
             entry/randomization).

          -  Glomerular filtration rate (GFR) >= 50 mL/min/1.73 m^2 according to the Chronic Kidney
             Disease Epidemiology Collaboration (CKD-EPI) equation. If not on target, the
             evaluation may be repeated once after at least 24 hours either according to the
             CKD-EPI equation or by 24 hour sampling. If the later result is within acceptable
             range, it may be used to fulfill the inclusion criteria instead.

          -  Patients must be able to swallow and tolerate oral medications and not have
             gastrointestinal illnesses that would preclude absorption of niraparib (e.g.
             uncontrolled nausea, vomiting, or diarrhea; malabsorption syndrome; ulcerative
             disease).

          -  Participants' life expectancy must be at least 3 months.

          -  Patients must be able to understand and willing to sign an informed consent.

          -  Patients must agree to not donate blood during the study or for 90 days after the last
             dose of study treatment.

          -  For the expansion phase only: For the expansion phase, patients with recurrent
             platinum sensitive disease must progress after treatment with a platinum doublet, and
             patients with BRCA mutations must progress after maintenance therapy with PARP
             inhibitor.

          -  For the expansion phase only: For the expansion phase, patients must have measurable
             disease accessible for biopsy.

          -  For the expansion phase only: Archival specimens from the time of primary or
             recurrence diagnosis.

        Exclusion Criteria:

          -  Patients must not be simultaneously enrolled in an interventional clinical trial.

          -  Patients with endometrial cancer may not have carcinosarcoma.

          -  Patients who have recurrences that are amenable to potentially curative treatment with
             radiation therapy or surgery.

          -  Patients who have a history of other malignancies except for basal cell or squamous
             cell skin cancer, in situ cervical cancer, unless they have been disease-free for at
             least three years. Patients may have dual primaries of ovarian and endometrial cancer,
             superficial bladder, or localized prostate cancer.

          -  Patients who have had prior chemotherapy, biological therapy, radiation therapy,
             androgens, thalidomide, immunotherapy, other anticancer agents, and any
             investigational agents within 28 days of starting study treatment or autologous
             transplant less than 3 months before start of treatment unless evidence of progression
             since last treatment (not including palliative radiotherapy at focal sites) or within
             a time interval less than at least 5 half-lives of the investigational agent,
             whichever is longer, prior to the first scheduled day of dosing in this study.

          -  Patients who have had major surgery within 3 weeks prior to entry into the study or be
             recovering from any effects of surgery. Patients may not have had minor surgery within
             2 weeks or open biopsy less than 7 days prior to entry into the study.

          -  Patients with known hypersensitivity to niraparib or copanlisib or any of their
             excipients. Patients may not have history of hypersensitivity to drugs with a similar
             chemical structure or class to niraparib or copanlisib.

          -  Patients who have experienced intolerable adverse events per treating investigator due
             to other PARP inhibitors or PI3K-pathway inhibitors.

          -  Patients with known history of cancer involvement of the central nervous system. A
             scan to confirm absence of brain metastasis is not required.

          -  Congestive heart failure > New York Heart Association (NYHA) class 2.

          -  Unstable angina (angina symptoms at rest), new-onset angina (begun within the last 3
             months), myocardial infarction less than 6 months before start of test drug
             Uncontrolled hypertension (despite optimal medical management).

          -  Arterial or venous thrombotic or embolic events such as cerebrovascular accident
             (including transient ischemic attacks), deep vein thrombosis or pulmonary embolism
             within 3 months before the start of study treatment.

          -  Non-healing wound, ulcer, or bone fracture.

          -  Glycosylated hemoglobin (HbA1c) > 8.5% at screening.

          -  Known history of human immunodeficiency virus (HIV) infection. All patients must be
             screened for HIV up to 28 days prior to study drug start using a blood test for HIV
             according to local regulations.

          -  Hepatitis B (HBV) or hepatitis C (HCV). All patients must be screened for HBV and HCV
             up to 28 days prior to study drug start using the routine hepatitis virus laboratorial
             panel. Patients positive for hepatitis B surface antigen (HBsAg) or hepatitis B virus
             core antibody (HBcAb) will be eligible if they are negative for HBV-deoxyribonucleic
             acid (DNA); patients positive for anti-HCV antibody will be eligible if they are
             negative for HCV-ribonucleic acid (RNA).

          -  Patients with seizure disorder requiring medication.

          -  Proteinuria of >= Common Terminology Criteria for Adverse Events (CTCAE) (version 5.0)
             grade 3 as assessed by a 24 hour total urine protein (estimated by urine protein :
             creatinine ratio 3.5 on a random urine sample).

          -  History or concurrent condition of interstitial lung disease of any severity and/or
             severely impaired lung function (as judged by the investigator).

          -  Concurrent diagnosis of pheochromocytoma.

          -  Unresolved toxicity higher than CTCAE (version 5.0) grade 1 attributed to any prior
             therapy/procedure, excluding alopecia, peripheral neuropathy, and bone marrow
             parameters.

          -  Any illness or medical conditions that are unstable or could jeopardize the safety of
             patients and their compliance in the study.

          -  As judged by the investigator, the patient is unsuitable to participate in the study
             and the patient is unlikely to comply with study procedures, restrictions, and
             requirements.

          -  Active, clinically serious infections > CTCAE (version 5.0) grade 2.

          -  History of, or current autoimmune disease, including autoimmune thyroid disease.

          -  Cytomegalovirus (CMV) infection. Patients who are CMV polymerase chain reaction (PCR)
             positive at baseline will not be eligible.

          -  Patients with evidence or history of bleeding diathesis. Any hemorrhage or bleeding
             event >= CTCAE (version 5.0) grade 3 within 4 weeks prior to the start of the study
             medication.

          -  Pregnant or breast feeding women. Women of childbearing potential must have a serum
             pregnancy test performed a maximum of 7 days before start of treatment, and a negative
             result must be documented before start of treatment.

          -  History of having received an allogenic bone marrow or organ transplant.

          -  Patient has any known history of myelodysplastic syndrome (MDS) or acute myeloid
             leukemia (AML).

          -  Patients has had radiation therapy encompassing > 20% of the bone marrow within 2
             weeks or any radiation therapy within 1 week prior to day 1 of protocol therapy.

          -  Patient must not have received colony stimulating factors (e.g., granulocyte colony
             stimulating factor, granulocyte macrophage colony stimulating factor, or recombinant
             erythropoietin) within 4 weeks prior initiating protocol therapy.

          -  EXCLUDED PRIOR THERAPIES AND MEDICATIONS: - Myeloid growth factors less than 14 days
             before start of treatment - Blood or platelet transfusion less than 7 days before
             start of treatment - Ongoing systemic corticosteroid therapy at a daily dose higher
             than 15 mg prednisone or equivalent. Previous corticosteroid therapy must be stopped
             or reduced to the allowed dose 7 days before performing the screening computed
             tomography (CT)/magnetic resonance imaging (MRI) and again prior to the first study
             drug administration. If a patient is on chronic corticosteroid therapy,
             corticosteroids should be de-escalated to the maximum allowed dose before the
             screening. Patients may use topical or inhaled corticosteroids. - Anti-arrhythmic
             therapy (beta blockers or digoxin are permitted) - Use of strong inhibitors and
             inducers of CYP3A4 is prohibited from day -14 of cycle 1 until the end of study visit.
             Copanlisib is primarily metabolized by CYP3A4. Therefore concomitant use of strong
             inhibitors of CYP3A4(e.g., ketoconazole, itraconazole, clarithromycin, ritonavir,
             indinavir, nelfinavir and saquinavir), and inducers of CYP3A (e.g. rifampin,
             phenytoin, carbamazepine, phenobarbital, St. John's wort) are not permitted from day
             -14 of cycle 1 until the end of study visit.

          -  For expansion phase only: Lack of accessible tumor for biopsy.