Phase 2 Study of MPDL3280A Combined With CDX-1401 in NY-ESO 1 (+) IIIB, IV or Recurrent Non-Small Cell Lung Cancer

Inclusion Criteria: A. Signed Informed Consent B. Ability to comply with the protocol C. Age ≥18 years D. Histologically or cytologically documented, locally advanced or metastatic (i.e., Stage IIIB not eligible for definitive chemoradiotherapy, Stage IV, or recurrent) NSCLC (per the American Joint Committee /AJCC staging system) E. Measurable disease, as defined by RECIST v1.1. Previously irradiated lesions can be counted as target lesions if clearly progressing after radiation. F. Chemotherapy naive and treated patients will be eligible, with no limit on number of prior therapies. Patients with NSCLC known to harbor an ALK rearrangement, or EGFR mutation known to be sensitive to FDA approved tyrosine kinase inhibitors (TKI), are only eligible after experiencing disease progression (during or after treatment) or intolerance to an FDA approved EGFR TKI or ALK TKI, respectively. G. Positive NY-ESO-1 expression by RT-PCR and/or IHC will be required for entry, as determined by analysis at the trial central laboratory. H. At least one tumor amenable to excisional, core or forceps (transbronchial) biopsy. Patients must be willing to undergo tumor biopsies before starting therapy and after the 3rd CDX-1401 injection. Additionally, the first 12 patients enrolled must consent to a third tumor biopsy to be performed after the 3rd MPDL3280A infusion. I. ECOG performance status of 0 to 2 J. For female patients of childbearing potential and male patients with partners of childbearing potential, agreement (by patient and/or partner) to use a highly effective form(s) of contraception (i.e., one that results in a low failure rate \[\<1% per year\] when used consistently and correctly) and to continue its use for 6 months after the last dose of trial therapy. Highly effective contraception is one with a failure rate of \<0.1%. Birth control pills on their own do not achieve that rate. K. Adequate hematologic and end-organ function, defined by the following laboratory results obtained within 14 days prior to the first study treatment: * ANC ≥1500 cells/μL (without granulocyte colony-stimulating factor support within 2 weeks prior to Cycle 1, Day 1) * Platelet count ≥100,000/μL (without transfusion within 2 weeks prior to Cycle 1, Day 1) * Hemoglobin ≥9.0 g/dL (Patients may be transfused to meet this criterion) * AST, ALT, and ALP ≤2.5 xULN, with the following exceptions: Patients with documented liver metastases: AST and/or ALT≤5 x ULN; Patients with documented liver or bone metastases: ALP ≤5 x ULN * Serum bilirubin ≤1.5 xULN (Patients with known Gilbert disease who have serum bilirubin level ≤3 xULN may be enrolled) * INR and aPTT≤1.5 x ULN (This applies only to patients who are not receiving therapeutic anticoagulation; patients receiving therapeutic anticoagulation should be on a stable dose) * Serum creatinine ≤1.5 xULN or creatinine clearance ≥50 mL/min Exclusion Criteria: A. Has an active autoimmune disease requiring systemic treatment within the past 3 months or a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids or immunosuppressive agents. Subjects with vitiligo or resolved childhood asthma/atopy would be an exception to this rule. Subjects who require intermittent use of inhaled steroids or local steroid injections would not be excluded from the study. Subjects with hypothyroidism stable on hormone replacement, or psoriasis not requiring systemic therapy (within the past 3 years) will not be excluded from the study. B. Generalized dermatologic conditions (such as allergic reactions, infection, edema, or scarring) that will not allow for study drug administration at a site of normal skin or evaluation of localized adverse events. C. Symptomatic or untreated CNS metastases. Patients with a history of treated asymptomatic CNS metastases are eligible, provided they meet all of the following criteria: No evidence of interim progression between the completion of CNS-directed therapy and the start of trial therapy. No ongoing requirement for dexamethasone as therapy for CNS disease; anticonvulsants at a stable dose are allowed. Completed stereotactic radiation at least 1 week prior to Cycle 1, Day 1 or whole-brain radiation at least 2 weeks prior to Cycle 1, Day 1 D. Treatment with systemic immunosuppressive medications (including but not limited to, prednisone at doses \> 10 mg (or equivalent dose of other corticosteroids), cyclophosphamide, tacrolimus, sirolimus, azathioprine, methotrexate, thalidomide, and antitumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to CDX-1401 administration (Inhaled or topically applied steroids, and acute and chronic standard-dose NSAIDs are permitted. Replacement steroids are also permitted). E. Any approved anti-cancer therapy, including chemotherapy, or hormonal therapy within 3 weeks prior to initiation of study treatment; the following exceptions are allowed: * Hormone-replacement therapy or oral contraceptives * TKIs approved for treatment of NSCLC discontinued \> 7 days prior to Cycle 1, Day 1. The baseline scan must be obtained after discontinuation of prior TKIs. F. Treatment with any other investigational agent or participation in another clinical trial with therapeutic intent within 28 days prior to enrollment; the following exceptions are allowed: \- Unapproved/ experimental TKIs discontinued 14 days prior to Cycle 1, Day 1 G. Known infection with HIV, HBV or HCV. Patients with prior exposure to hepatitis, but no evidence of active or chronic infection, may be eligible. H. Active systemic infection requiring systemic antibiotic treatment within 72 hours prior to first dose of study treatment I. Uncontrolled intercurrent illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, or psychiatric illness/social situations that would limit compliance with study requirements J. Women who are pregnant or lactating. K. Any underlying medical condition that in the Principal Investigator's opinion will make the administration of study drug hazardous to the patient or would obscure the interpretation of adverse events. L. Previous administration of vaccine therapy targeting NY-ESO-1 M. Prior treatment with immune checkpoint blockade therapies, including anti-CTLA4, anti-PD-1, and anti-PD-L1 therapeutic antibodies