Inclusion Criteria:

          -  Cohort 1 only: Patients must have pathologically or cytologically confirmed adenoid
             cystic carcinoma. Cancers arising from non-salivary gland primary sites are allowed.

          -  Patients must have pathologically or cytologically confirmed salivary gland cancer of
             any histology except for adenoid cystic carcinoma.

          -  Patients must have recurrent and/or metastatic disease not amenable to potentially
             curative surgery or radiotherapy.

          -  At least 2 weeks must have elapsed since the end of prior systemic treatment and/or 4
             weeks since completion of radiotherapy with resolution of all treatment-related
             toxicity to NCI CTCAE Version 4.0 grade ≤1 (or tolerable grade 2) or back to baseline
             (except for alopecia, lymphopenia, or hypothyroidism) prior to starting study drug
             treatment. Any number of prior therapies for recurrent/metastatic salivary gland
             cancer are allowed.

        NOTE: Patients previously treated with hormonal therapies (e.g., drugs targeting the
        androgen receptor) may continue these drugs prior to trials enrollment and concomitantly
        with study therapy.

          -  Patients must have RECIST v1.1 measurable disease.

          -  Cohort 1 and acinic cell carcinoma patients in Cohort 2 only: Patients must have
             documentation of a new or progressive lesion on a radiologic imaging study performed
             within 6 months prior to study enrollment (progression of disease over any interval is
             allowed) and/or new/worsening disease related symptoms within 6 months prior to study
             enrollment. Note: This assessment will be performed by the treating investigator.
             Evidence of progression by RECIST criteria is not required.

          -  Age ≥ 18 years.

          -  ECOG performance status 0 or 1 (or Karnofsky ≥ 70%).

          -  Patients must have tissue from the primary tumor or metastases available for
             correlative studies. Either a paraffin block or at least 20 unstained slides are
             acceptable (30 unstained slides would be ideal). (If less than twenty unstained slides
             are available and a paraffin bloc is not available, the patient may be able to
             participate at the discretion of the investigator.)

          -  Patients must agree to undergo two research biopsies of malignant lesions. Tumor
             tissue obtained prior to study consent or treatment as part of standard of care can
             also be submitted in lieu of performance of the first pre-treatment biopsy, if the
             Principal Investigator deems it to be of sufficient quantity/quality/timeliness.
             Patients may be exempt from biopsy if 1) the investigator or person performing the
             biopsy judges that no tumor is accessible for biopsy, 2) the investigator or person
             performing the biopsy feels that the biopsy poses too great of a risk to the patient,
             or 3) the patient's platelet count is <100,000/mcl or he/she cannot be safely removed
             from anti-coagulation therapy (if the anti-coagulation therapy needs to be temporarily
             held for the biopsy procedure). If the only tumor accessible for biopsy is also the
             only lesion that can be used for RECIST v1.1 response evaluation, then the patient may
             be exempt from biopsy. If the investigator deems a second research biopsy to be high
             risk after a patient has completed the first research biopsy, the patient may be
             exempt from the second biopsy.

          -  Screening laboratory values must meet the following criteria:

               -  WBC ≥ 2000/μL

               -  Neutrophils ≥ 1500/μL

               -  Platelets ≥ 100 x10^3/μL

               -  Hemoglobin > 9.0 g/dL

               -  AST/ALT ≤ 3 x ULN

               -  Total Bilirubin ≤ 1.5 x ULN (except subjects with Gilbert Syndrome, who can have
                  total bilirubin < 3.0 mg/dL)

               -  Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 40 mL/min (if using
                  the Cockcroft-Gault formula below):

        Female CrCl = (140 - age in years) x weight in kg x 0.85 72 x serum creatinine in mg/dL

        Male CrCl = (140 - age in years) x weight in kg x 1.00 72 x serum creatinine in mg/dL

          -  Women of childbearing potential (WOCBP)* must use appropriate method(s) of
             contraception. WOCBP should use an adequate method to avoid pregnancy for 23 weeks (30
             days plus the time required for nivolumab to undergo five half-lives) after the last
             dose of investigational drug.

               -  WOCBP is defined as any female who has experienced menarche and who has not
                  undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who
                  is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea
                  in a woman over 45 in the absence of other biological or physiological causes.

        Women who are not of childbearing potential are not required to use contraception.

          -  Women of childbearing potential must have a negative serum or urine pregnancy test
             upon study entry.

          -  Men who are sexually active with women of child bearing potential must use adequate
             contraception upon study entry until 31 weeks after the last dose of study treatment.
             Men who are surgically sterile or azoospermic do not require contraception.

        Exclusion Criteria:

          -  Symptomatic metastatic brain or leptomeningeal tumors (asymptomatic or treated
             metastatic brain or leptomeningeal tumors are allowed).

          -  Current or prior use of immunosuppressive doses of systemic corticosteroids (>10
             mg/day prednisone equivalents) or other immunosuppressive medications within 2 weeks
             of study drug administration. NOTE: Subjects are permitted to use topical, ocular,
             intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic
             absorption). Physiologic replacement doses of systemic corticosteroids are permitted,
             even if >10 mg/day prednisone equivalents. A brief course of corticosteroids for
             prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions
             (eg, delayed-type hypersensitivity reaction caused by contact allergen) is permitted.

          -  Active, known, or suspected autoimmune disease within the past 2 years. NOTE: 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

          -  Patients should be excluded if they have had prior systemic treatment with an
             anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug
             specifically targeting T-cell costimulation or immune checkpoint pathways.

          -  Patients should be excluded if they have a known history of testing positive for
             hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus antibody (HCV
             antibody) indicating acute or chronic infection (those with treated hepatitis B or C
             infection and a negative viral load prior to study entry would be eligible)

          -  Patients should be excluded if they have a known history of testing positive for human
             immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS)

          -  History of allergy to study drug components.

          -  History of severe hypersensitivity reaction to any monoclonal antibody.

          -  Women who are pregnant or breast-feeding.