Inclusion Criteria:

          -  Male or female ≥ 18 years of age at time of consent.

          -  Histologically-confirmed malignant mesothelioma not amenable to curative surgery and
             who have received at least one pemetrexed-containing chemotherapy regimen.

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1.

          -  total bilirubin < 1.5 mg/dL (25.65 μmol/L) OR direct bilirubin ≤ ULN for subjects with
             total bilirubin levels > 1.5 mg/dL (except subject with Gilbert's Syndrome, who can
             have total bilirubin < 3.0 mg/dl)

          -  aspartate aminotransferase (AST) ≤ 3 × ULN or ≤ 5 × ULN for subjects with known
             hepatic metastases

          -  alanine aminotransferase (ALT) ≤ 3 × ULN or ≤ 5 × ULN for subjects with known hepatic
             metastases

          -  hemoglobin ≥ 8 g/dL, subjects requiring transfusion will not be eligible to start
             study

          -  absolute neutrophil count (ANC) ≥ 1.5 × 109/L

          -  platelet count ≥ 100 × 109/L

          -  serum creatinine ≤1.5 times the ULN, or creatinine clearance (measured via 24-hour
             urine collection) ≥40 mL/minute (that is, if serum creatinine is >1.5 times the ULN, a
             24-hour urine collection to calculate creatinine clearance must be performed)

          -  subject's urinary protein is ≤1+ on dipstick or routine urinalysis (UA; if urine
             dipstick or routine analysis is ≥2+, a 24-hour urine collection for protein must
             demonstrate <1000 mg of protein in 24 hours to allow participation in this protocol)

          -  INR < 1.5, and a partial thromboplastin time (PTT) (PTT/aPTT) < 1.5 x ULN (unless
             receiving anticoagulant therapy)

          -  Subjects on full-dose anticoagulation must be on a stable dose (minimum duration 14
             days) of oral anticoagulant or low molecular weight heparin (LMWH). Patients receiving
             warfarin must be switched to low molecular weight heparin and have achieved stable
             coagulation profile prior to first dose of protocol therapy. For heparin and LMWH
             there should be no active bleeding (that is, no bleeding within 14 days prior to first
             dose of protocol therapy) or pathological condition present that carries a high risk
             of bleeding (for example, tumor involving major vessels or known varices).

          -  Subjects must be willing to undergo a CT-guided biopsy (i.e., image-guided
             percutaneous lung biopsy) to obtain tumor tissue within 28 days before initiation of
             treatment and after 4 cycles (8 weeks) of treatment.

          -  Women of childbearing potential (WOCP) must be willing to use birth control as
             outlined in the protocol.

          -  Men who are not surgically or medically sterile must agree to use contraception as
             outlined in the protocol.

          -  Measurable disease, defined as at least 1 tumor that fulfills the criteria for a
             target lesion according to modified RECIST 1.1 criteria, and obtained by imaging
             within 28 days prior to study registration.

          -  Prior intracavity cytotoxic or sclerosing agents (including bleomycin) is acceptable.

          -  Radiation therapy must be completed > 28 days before study registration, and the
             measurable disease must be outside of the radiation port.

          -  Pemetrexed-containing chemotherapy must be completed > 28 days before study
             registration.

          -  Must provide written informed consent and HIPAA authorization approved by an
             Institutional Review Board (IRB). NOTE: HIPAA authorization may be included in the
             informed consent or obtained separately.

          -  All previous toxicity resolved to Grade 1 or less.

        Exclusion Criteria:

          -  Any Grade 3-4 GI bleeding within 3 months prior to study registration.

          -  History of deep vein thrombosis, pulmonary embolism, or any other significant
             thromboembolism (venous port or catheter thrombosis or superficial venous thrombosis
             are not considered "significant") during the 3 months prior to study registration.

          -  Any arterial thromboembolic events, including but not limited to myocardial
             infarction, transient ischemic attack, cerebrovascular accident, or unstable angina,
             within 6 months prior to study registration.

          -  Cirrhosis at a level of Child-Pugh B (or worse), or cirrhosis (any degree) with a
             history of hepatic encephalopathy, or clinically meaningful ascites resulting from
             cirrhosis. Clinically meaningful ascites is defined as ascites from cirrhosis
             requiring diuretics or paracentesis.

          -  Uncontrolled or poorly-controlled hypertension (>160 mmHg systolic or > 100 mmHg
             diastolic for >4 weeks) despite standard medical management.

          -  Prior history of GI perforation/fistula (within 6 months of study registration) or
             risk factors for perforation.

          -  Serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to study
             registration.

          -  Active brain metastases or carcinomatous meningitis. Subjects with neurological
             symptoms must undergo a head CT scan or brain MRI to exclude brain metastasis.
             Subjects with previously treated brain metastases may participate provided they are
             stable (without evidence of progression by imaging for at least four weeks prior to
             the first dose of study drugs and any neurologic symptoms have returned to baseline),
             have no evidence of new or enlarging brain metastases, and are not using steroids for
             at least 14 days prior to study registration. This exception does not include
             carcinomatous meningitis, which is excluded regardless of clinical stability.

          -  Major surgery within 28 days prior to study registration

          -  Subcutaneous venous access device placement within 7 days prior to study registration.

          -  Elective or planned major surgery to be performed during the course of the clinical
             trial.

          -  Is receiving chronic antiplatelet therapy, including aspirin, nonsteroidal
             anti-inflammatory drugs (NSAIDs, including ibuprofen, naproxen, and others),
             dipyridamole or clopidogrel, or similar agents. Once-daily aspirin use (maximum dose
             325 mg/day) is permitted.

          -  Known history of testing positive for human immunodeficiency virus (HIV) or known
             acquired immunodeficiency syndrome (AIDS). NOTE: HIV testing is not required.

          -  Known history of testing positive for hepatitis B virus surface antigen (HBV sAg) or
             hepatitis C virus ribonucleic acid (HCV antibody), indicating acute or chronic
             infection. NOTE: Hepatitis B and Hepatitis C testing is not required.

          -  Condition requiring systemic treatment with either corticosteroids (> 10 mg daily
             prednisone equivalents) or other immunosuppressive medications within 14 days of study
             drug administration. Inhaled or topical steroids and adrenal replacement doses > 10 mg
             daily prednisone equivalents are permitted in the absence of active autoimmune
             disease.

          -  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). Subjects are permitted to enroll if they have vitiligo, type I diabetes
             mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone
             replacement, psoriasis not requiring systemic treatment, or conditions not expected to
             recur in the absence of an external trigger.

             o NOTE: 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.

          -  Received a live vaccine within 30 days prior to the first dose of trial treatment.
             Examples of live vaccines include, but are not limited to: measles, mumps, rubella,
             chicken pox, yellow fever, rabies, BCG, and typhoid (oral) vaccine. Seasonal influenza
             vaccines for injection are generally killed virus vaccines and are allowed; however,
             intranasal influenza vaccines (e.g. Flu-Mist®) are live attenuated vaccines and are
             not allowed.

          -  History of interstitial lung disease or active, non-infectious pneumonitis.

          -  Female subject is pregnant or breast-feeding.

               -  NOTE: Women of childbearing potential (WOCP) must have a negative pregnancy test
                  (either serum β-HCG with a sensitivity of 50 mIU/ml or urine dipstick within 24
                  hours of study registration).

               -  NOTE: Women are not considered to be of childbearing potential if they meet at
                  least one of the following: 1) surgically sterilized, or 2) postmenopausal (a
                  woman who is ≥45 years of age and has not had menses for greater than 1 year), or
                  3) not heterosexually active for the duration of the study. See section 5.6.2.

          -  Major blood vessel invasion or significant intratumor cavitation.

          -  If they experience hemoptysis (defined as bright red blood or ≥ ½ teaspoon) within 2
             months prior to first dose of protocol therapy or with radiographic evidence of
             intratumor cavitation or has radiologically documented evidence of major blood vessel
             invasion or encasement by cancer.

          -  Any condition that, in the opinion of the investigator, might jeopardize the safety of
             the subject or interfere with protocol compliance.

          -  Any mental or medical condition that prevents the subject from giving informed consent
             or participating in the trial.

          -  Any pathological condition that carries a high risk of bleeding (for example, tumor
             involving major vessels or known varices.)

          -  Known hypersensitivity to nivolumab or ramucirumab or any of their components.

          -  Known history of active tuberculosis.

          -  Received prior therapy with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CD137, or
             anti-Cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (including
             ipilimumab or any other antibody or drug specifically targeting T-cell co-stimulation
             or checkpoint pathways).

          -  No prior malignancy is allowed except for adequately treated basal cell or squamous
             cell skin cancer, in situ cervical cancer, localized Gleason ≤ grade 7 prostate
             cancers. Subjects with other solid tumors treated curatively and without evidence of
             recurrence for at least 3 years prior to enrollment may be eligible for study after
             discussion with the sponsor-investigator

          -  Treatment with any investigational agent within 28 days prior to study registration.
             The subject must have recovered from the acute toxic effects of the regimen.