Inclusion Criteria

          1. Signed written informed consent granted prior to initiation of any study-specific
             procedures

          2. 18 years of age and older

          3. Histologically and/or cytologically documented diagnosis of a selected tumor type that
             is locally advanced, inoperable, metastatic or recurrent (including but not restricted
             to breast cancer, TNBC [triple negative]; HR-positive [HR+]/HER2-negative [HER2-] or
             endometrial cancer)

          4. Documented AKT genetic alterations or known actionable PIK3CA/PTEN mutations by
             genetic testing

             • Subjects with tumors with PTEN null/PTEN loss-of-function mutations are not eligible

          5. For combination arms; subjects should be eligible for paclitaxel or fulvestrant
             therapy as per Investigator assessment

          6. Failure to respond to standard systemic therapy, or for whom standard or curative
             systemic therapy does not exist or is not tolerable

               -  Subjects in single agent arm (with AKT genetic alterations) and subjects in dose
                  escalation cohorts of combination therapy arms should have at least one line of
                  standard systemic therapy

               -  Subjects in single agent arm (with PIK3CA/PTEN actionable mutations) and subjects
                  in the expansion cohorts of combination therapy arms should have no more than 3
                  prior systemic regimens for the advanced disease

               -  Neoadjuvant and adjuvant chemotherapy are considered one regimen if they are a
                  continuation of the same regimen with interval debulking surgery

               -  If the subject is refractory or has disease progression within 6 months after
                  completion of the adjuvant treatment, then the adjuvant treatment should be
                  considered as the line of treatment rather than an adjuvant therapy.

               -  Endocrine (hormonal) therapy does not count toward total lines of therapy

               -  Maintenance therapy is considered part of the preceding regimen if one or more of
                  the same drugs are continued

          7. Has at least one measurable target lesion according to RECIST v. 1.1

          8. Eastern Cooperative Oncology Group (ECOG) performance status (PS) ≤ 1

          9. Adequate organ function as indicated by the following laboratory values. (All
             laboratory tests must be obtained within 14 days prior to the first dose of study
             treatment):

               1. Hematological

                    -  Absolute neutrophil count (ANC) ≥ 1.5 x 10⁹/L

                    -  Platelet count (Plt) ≥ 100 x 10⁹/L

                    -  Hemoglobin (Hb) ≥ 9 g/dL

                    -  International normalized ratio (INR) 0.8 to upper limit of normal (ULN) or ≤
                       3 for subjects receiving anticoagulant therapy such as Coumadin or heparin

               2. Renal

                    -  Serum creatinine ≤ 1.5 x ULN or calculated creatinine clearance ≥ 60
                       mL/min/1.73 m2 for subjects with serum creatinine levels > 1.5 x
                       institutional ULN

               3. Hepatic

                    -  Total bilirubin ≤ 1.5 x ULN

                    -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 x
                       ULN or ≤ 5 x ULN for subjects with known liver metastases

               4. Metabolic

                    -  Glycated hemoglobin (HbA1c) ≤ 8% (≤ 64 mmol/mol)

         10. If a subject is currently receiving bisphosphonates or any other drug for treatment of
             osteoporosis, treatment-induced bone loss and metastases to bone, the subject must
             have received the bisphosphonates for at least four weeks prior to the first dose of
             study treatment

             • Initiation of bisphosphonates or similar agents during the study may be allowed
             provided the subject completes the first cycle of treatment without any dose limiting
             toxicity (DLT) and the Investigator rules out tumor progression

         11. Male or female subjects of child-producing potential must agree to use adequate
             contraception, including double-barrier contraceptive measures, oral contraception, or
             avoidance of intercourse during the study and for 90 days after the last dose of study
             treatment

         12. Women of childbearing potential must have a negative serum pregnancy test. "Women of
             childbearing potential" is defined as sexually mature women who have not undergone a
             hysterectomy or who have not been naturally postmenopausal for at least 12 consecutive
             months prior to the first dose of study treatment

        Exclusion Criteria

          1. Anti-cancer therapy, such as chemotherapy, immunotherapy, hormonal therapy, targeted
             therapy, or investigational agents within five half-lives or four weeks, whichever is
             shorter, prior to administration of the first dose of study treatment

               -  To be eligible for study treatment, toxicity from prior treatment(s) must recover
                  to Grade ≤ 1, except for alopecia

               -  Concurrent systemic high-dose corticosteroids (in dosing exceeding 10 mg QD of
                  prednisone equivalent) when used intermittently in an antiemetic regimen, for
                  central nervous system (CNS) metastases management, or as a part of the
                  premedication regimen are allowed

          2. Radiation therapy within four weeks, or palliative radiation therapy within two weeks,
             prior to administration of the first dose of study treatment

               -  To be eligible for study treatment, radiation therapy-related toxicity must
                  recover to Grade ≤ 1 prior to administration of the first dose of study treatment

               -  Concurrent palliative radiotherapy for local pain-control or prevention of
                  fracture (for known bone metastases) may be allowed provided the subject
                  completes the first cycle of treatment, does not meet criteria of progressive
                  disease, and treated lesions will not be included in the target/non-target lesion
                  assessment

          3. Major surgical procedure within four weeks prior to administration of the first dose
             of study treatment

             • To be eligible for the study treatment, all surgical wounds must be fully healed,
             and any surgery-related adverse events must recover to Grade ≤ 1.

          4. Unable or unwilling to swallow the complete daily dose of vevorisertib

          5. Previous treatment with

               -  AKT inhibitors (e.g., ARQ 092, MK-2206, GSK2141795, AZD5363; prior treatment with
                  PI3K or mTOR inhibitor are allowed)

               -  Prior taxane therapy for the advanced, metastatic disease (for subjects
                  considered for vevorisertib +paclitaxel combination arm only)

          6. Known prior allergic reaction to or severe intolerance of paclitaxel or fulvestrant.
             Intolerance is defined as a serious adverse event, a grade 3 or 4 AE per CTCAE v.4.03,
             or permanent treatment discontinuation

          7. History of Type 1 diabetes mellitus or Type 2 diabetes mellitus requiring regular
             medication (other than oral hypoglycemic agents) or fasting glucose ≥ 160 mg/dL at
             Screening visit

          8. Significant gastrointestinal disorder(s) that could, in the opinion of the
             Investigator, interfere with the absorption, metabolism, or excretion of vevorisertib
             (e.g., inflammatory bowel disease, Crohn's disease, ulcerative colitis, extensive
             gastric resection)

          9. Known untreated or active CNS metastases and/or carcinomatous meningitis

             • To be eligible for the study treatment, subjects must have stable disease ≥ 1 month,
             confirmed by magnetic resonance imaging (MRI) or computed tomography (CT) scan, and
             have CNS metastases well controlled by low-dose steroids, anti-epileptics, or other
             symptom-relieving medications

         10. History of myocardial infarction (MI) or New York Heart Association (NYHA) Class II-IV
             congestive heart failure within 6 months of the administration of the first dose of
             study treatment (MI occurring > 6 months of the first dose of study treatment will be
             permitted); Grade 2 or worse conduction defect (e.g., right or left bundle branch
             block)

         11. A heart rate corrected QT (QTc) interval ≥ 480 msec, using the Fridericia's formula
             QTcF

         12. Left ventricular ejection fraction (LVEF) <50% as determined by Multiple Gated
             Acquisition (MUGA) scan or echocardiogram (ECHO) in subjects who received prior
             treatment with anthracyclines

         13. Concurrent severe and/or uncontrolled illness not related to cancer and/or social
             situation that would limit compliance with study requirements, including but not
             limited to:

               -  Psychiatric illness, substance abuse

               -  Ongoing or active known infection, including human immunodeficiency virus (HIV)
                  infection, hepatitis B or C virus

               -  Significant pulmonary dysfunction, including pneumonitis, interstitial lung
                  disease (ILD), idiopathic pulmonary fibrosis, cystic fibrosis, severe COPD

               -  Peripheral neuropathy grade ≥2 (vevorisertib+paclitaxel combination arm)

               -  Bleeding diathesis, thrombocytopenia or coagulation disorders
                  (vevorisertib+fulvestrant combination arm)

               -  Thrombotic/coagulation disorders within 6 months prior to the first dose of study
                  treatment unless stable on anticoagulation for > 3 months

         14. Active or history of other malignancy other than the current cancer within 2 years of
             the first dose of study treatment, with the exception of carcinoma in-situ of the
             cervix, basal cell carcinoma, and superficial bladder tumors curatively treated

         15. Blood transfusion or administration of growth factors within 5 days prior to a blood
             draw being used to confirm eligibility

         16. Pregnant or breastfeeding