Inclusion Criteria:

          -  Participants who are at least 18 years of age on the day of signing informed consent
             will be enrolled in this study

          -  Women with histologically confirmed endometrial cancer, epithelial ovarian cancer,
             fallopian tube cancer or primary peritoneal cancer (all histological subtypes)

          -  Patients must have received prior treatment with a platinum containing regimen and may
             have received 1-3 prior regimens. Hormonal therapy and maintenance therapy (with
             bevacizumab or PARP inhibitor) does not count towards the number of prior treatments

          -  Patients with ovarian, fallopian tube or primary peritoneal cancer must be platinum
             resistant (progression < 6 months after completion of a platinum containing regimen)
             or not a candidate for further platinum treatment

          -  Patients with and without mismatch repair deficiency are allowed

          -  Prior PD-1/PD-L1 inhibitors are allowed as long as this was not the most recent
             regimen and treatment was not discontinued due to side effects/toxicity. Prior weekly
             paclitaxel is allowed as long as this was not the most recent regimen and treatment
             was not discontinued due to side effects/toxicity

          -  Prior targeted therapy targeting angiogenesis is allowed as long as treatment was not
             discontinued due to side effects/toxicity

          -  Have measurable disease based on RECIST 1.1. Lesions situated in a previously
             irradiated area are considered measurable if progression has been demonstrated in such
             lesions

          -  A female participant is eligible to participate if she is not pregnant, not
             breastfeeding, and at least one of the following conditions applies:

               -  Not a woman of childbearing potential (WOCBP) as defined OR

               -  A WOCBP who agrees to follow the contraceptive guidance during the treatment
                  period and for at least 4 months (corresponding to time needed to eliminate any
                  study treatment[s] MK and or any active comparator/combination) plus 30 days (a
                  menstruation cycle) after the last dose of study treatment

          -  The participant (or legally acceptable representative if applicable) provides written
             informed consent for the trial

          -  Have provided archival tumor tissue sample or newly obtained core or excisional biopsy
             of a tumor lesion not previously irradiated. Formalin-fixed, paraffin embedded (FFPE)
             tissue blocks are preferred to slides. Newly obtained biopsies are preferred to
             archived tissue.

          -  Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1.

          -  Absolute neutrophil count (ANC) >= 1500/uL (within 10 days prior to the start of study
             treatment)

          -  Platelets >= 100 000/uL (within 10 days prior to the start of study treatment)

          -  Hemoglobin >= 9.0 g/dL or >= 5.6 mmol/L (within 10 days prior to the start of study
             treatment)

               -  Criteria must be met without erythropoietin dependency and without packed red
                  blood cell (pRBC) transfusion within last 2 weeks

          -  Creatinine =< 1.5 x upper limit of normal (ULN) OR measured or calculated creatinine
             clearance (glomerular filtration rate [GFR] can also be used in place of creatinine or
             creatinine clearance [CrCl]) >= 30 mL/min for participant with creatinine levels > 1.5
             x institutional ULN (within 10 days prior to the start of study treatment)

               -  Creatinine clearance (CrCl) should be calculated per institutional standard

          -  Total bilirubin =< 1.5 x ULN OR direct bilirubin =< ULN for participants with total
             bilirubin levels > 1.5 x ULN (within 10 days prior to the start of study treatment)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT]) and
             alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase [SGPT]) =< 2.5 x
             ULN (=< 5 x ULN for participants with liver metastases) (within 10 days prior to the
             start of study treatment)

          -  International normalized ratio (INR) OR prothrombin time (PT) activated partial
             thromboplastin time (aPTT) =< 1.5 x ULN unless participant is receiving anticoagulant
             therapy as long as PT or aPTT is within therapeutic range of intended use of
             anticoagulants (within 10 days prior to the start of study treatment)

        Exclusion Criteria:

          -  A women of child birth potential (WOCBP) who has a positive urine pregnancy test
             within 72 hours prior to registration.

          -  Has received prior systemic anti-cancer therapy including investigational agents
             within 4 weeks or 5 half-lives whichever is shorter prior to study entry. Patients may
             not have received biologics, targeted therapy or immunotherapy for 4 weeks (6 weeks
             for nitrosoureas or mitomycin C). Hormonal therapy is not considered anti-neoplastic
             therapy.

          -  Has received prior radiotherapy within 3 weeks of start of study treatment.
             Participants must have recovered from all radiation-related toxicities, not require
             corticosteroids, and not have had radiation pneumonitis. A 2-week washout is permitted
             for palliative radiation (=< 2 weeks of radiotherapy) to non-central nervous system
             (CNS) disease

          -  Has received a live vaccine within 30 days prior to the first dose of study drug.
             Seasonal influenza vaccines for injection are generally killed virus vaccines and are
             allowed; however, intranasal influenza vaccines (e.g., FluMist) are live attenuated
             vaccines and are not allowed

          -  Is currently participating in or has participated in a study of an investigational
             agent or has used an investigational device within 4 weeks prior to the first dose of
             study treatment.

          -  Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy
             (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of
             immunosuppressive therapy within 7 days prior to the first dose of study drug

          -  Has a known additional malignancy that is progressing or has required active treatment
             within the past 3 years. Note: Participants with basal cell carcinoma of the skin,
             squamous cell carcinoma of the skin, or carcinoma in situ (e.g. breast carcinoma,
             cervical cancer in situ) that have undergone potentially curative therapy are not
             excluded

          -  Has known active CNS metastases and/or carcinomatous meningitis. Participants with
             previously treated brain metastases may participate provided they are radiologically
             stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging
             (note that the repeat imaging should be performed during study screening), clinically
             stable and without requirement of steroid treatment for at least 14 days prior to
             first dose of study treatment

          -  Has severe hypersensitivity (>= grade 3) to pembrolizumab, lenvatinib, or paclitaxel
             and/or any of its excipients. Note, patients with paclitaxel hypersensitivity are
             eligible if the if infusion can safely be completed with reduced infusion rates and
             premedication

          -  Has active autoimmune disease that has required systemic treatment in the past 2 years
             (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive
             drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid
             replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a
             form of systemic treatment

          -  Has a history of (non-infectious) pneumonitis that required steroids or has current
             pneumonitis

          -  Has a known history of hepatitis B (defined as hepatitis B surface antigen [HBsAg]
             reactive) or known active hepatitis C virus (defined as HCV RNA [qualitative] is
             detected) infection. Note: no testing for hepatitis B and hepatitis C is required
             unless the patient is at risk for, or has symptoms or signs of, hepatitis

          -  Has a known history of active TB (Bacillus Tuberculosis)

          -  Known human immunodeficiency virus (HIV)-positive patients on combination
             antiretroviral therapy are ineligible because of the potential for pharmacokinetic
             interactions with pembrolizumab, lenvatinib and paclitaxel. In addition, these
             patients are at increased risk of lethal infections when treated with
             marrow-suppressive therapy. Appropriate studies will be undertaken in patients
             receiving combination antiretroviral therapy when indicated

          -  Significant cardiovascular disease

          -  Major surgical procedure (e.g. laparotomy, bowel resection) 4 weeks prior to start of
             the study drug.

          -  Neuropathy grade > 1

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

          -  Active peptic ulcer disease, inflammatory bowel disease, ulcerative colitis, or other
             gastrointestinal condition with increased risk of perforation; history of abdominal
             fistula, gastrointestinal perforation, or intra-abdominal abscess within 4 weeks prior
             to administration of first dose of study drug

          -  Patients who have a history of a small or large bowel obstruction within 2 weeks of
             screening or who have and active partial small bowel obstruction or percutaneous
             endoscopic gastrostomy (PEG)-tube

          -  Serious/active infection or infection requiring parenteral antibiotics

          -  Significant arterial or venous thromboembolic disease or vascular disorders within 6
             months prior to administration of first dose of study drug, including by not limited
             to:

               -  Cerebrovascular accident (CVA) or transient ischemic attack (TIA)

               -  Peripheral ischemia > grade 2 (per Common Terminology Criteria for Adverse Events
                  [CTCAE] version 4.03)

               -  Arterial thrombotic event

               -  Patients who have a history of deep vein thrombosis or pulmonary embolus < 3
                  months prior to enrollment must be excluded. However, if patients are on
                  therapeutic anticoagulation, and the deep vein thrombosis (DVT)/pulmonary
                  embolism (PE) has been clinically stable or improved on anticoagulation for > 3
                  months, patients are eligible.

               -  Patients with a history of clotting disorders must be excluded

          -  Significant bleeding disorders within 6 months prior to administration of first dose
             of study drug

          -  Inability to swallow pills, malabsorption syndrome or gastrointestinal disease that
             severely affects the absorption of study drugs, major resection of the stomach or
             small bowel, or gastric bypass procedure

          -  Has a history or current evidence of any condition, therapy, or laboratory abnormality
             that might confound the results of the study, interfere with the subject's
             participation for the full duration of the study, or is not in the best interest of
             the subject to participate, in the opinion of the treating investigator

          -  Has known psychiatric or substance abuse disorders that would interfere with
             cooperation with the requirements of the trial

          -  Is pregnant or breastfeeding, or expecting to conceive within the projected duration
             of the study, starting with the screening visit through 120 days after the last dose
             of trial treatment. Pregnant women are excluded from this study because the potential
             for teratogenic or abortifacient effects. Because there is an unknown but potential
             risk for adverse events in nursing infants secondary to treatment of the mother with
             lenvatinib, breastfeeding should be discontinued if the mother is treated lenvatinib.
             These potential risks may also apply to other agents used in this study