Inclusion Criteria:

          -  Patients must have a histologically confirmed diagnosis of metastatic invasive triple
             negative breast cancer.

          -  Estrogen receptor (ER) and progesterone receptor (PR) less than Allred score of 3 OR
             less than 1% positive staining cells in the invasive component of the tumor.

          -  HER2 negative by fluorescence in situ hybridization (FISH) or immunohistochemistry
             (IHC) staining 0 or 1+.

          -  PD-L1 negative by any Food and Drug Administration (FDA) approved test.

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             nonnodal lesions and short axis for nodal lesions) as >= 20 mm (>= 2 cm) by chest
             x-ray or as >= 10 mm (>= 1 cm) with computed tomography (CT) scan, magnetic resonance
             imaging (MRI), or calipers by clinical exam.

          -  A tumor specimen obtained from relapsed metastatic or locally advanced disease (if
             applicable) must be submitted. Acceptable samples include core needle biopsies for
             deep tumor tissue (minimum 4 cores) or excisional, incisional, punch, or forceps
             biopsies for cutaneous, subcutaneous, or mucosal lesions. Formalin-fixed,
             paraffin-embedded (FFPE) tumor specimens in paraffin blocks are preferred. Fine-needle
             aspiration, brushing, cell pellet from pleural effusion, bone metastases, and lavage
             samples are not acceptable.

          -  No prior therapy for metastatic TNBC. Patients who have received taxane-based adjuvant
             therapy are required to have a disease-free interval of at least 12 months after
             completion of taxane therapy.

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 60%).

          -  Body weight > 30 kg.

          -  Must have a life expectancy of at least 12 weeks.

          -  Absolute neutrophil count >= 1,500/mcL.

          -  Platelets >= 100,000/mcL.

          -  Hemoglobin >= 9.0 g/dL.

          -  Serum bilirubin =< 1.5 x institutional upper limit of normal.

          -  Aspartate aminotransferase (AST) serum glutamic oxaloacetic transaminase
             (SGOT)/alanine aminotransferase (ALT) serum glutamic pyruvic transaminase (SGPT) =<
             2.5 x institutional upper limit of normal unless liver metastases are present, in
             which case it must be =< 5 x institutional upper limit of normal.

          -  Measured creatinine clearance > 40 mL/min.

          -  Calculated creatinine clearance > 40 mL/min by the Cockcroft-Gault formula or by
             24-hour urine collection for determination of creatinine clearance.

          -  Evidence of post-menopausal status or negative urinary or serum pregnancy test for
             female pre-menopausal patients. Women will be considered post-menopausal if they have
             been amenorrheic for 12 months without an alternative medical cause. The following
             age-specific requirements apply:

               -  Women < 50 years of age would be considered post-menopausal if they have been
                  amenorrheic for 12 months or more following cessation of exogenous hormonal
                  treatments and if they have luteinizing hormone and follicle-stimulating hormone
                  levels in the post-menopausal range for the institution or underwent surgical
                  sterilization (bilateral oophorectomy or hysterectomy).

               -  Women >= 50 years of age would be considered post-menopausal if they have been
                  amenorrheic for 12 months or more following cessation of all exogenous hormonal
                  treatments, had radiation-induced menopause with last menses > 1 year ago, had
                  chemotherapy-induced menopause with last menses > 1 year ago, or underwent
                  surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or
                  hysterectomy).

          -  The effects of durvalumab (MEDI4736) and neoantigen vaccine on the developing human
             fetus are unknown. For this reason and because these agents may be teratogenic, women
             of child-bearing potential must agree to use adequate contraception (hormonal or
             barrier method of birth control; abstinence) prior to study entry, for the duration of
             study participation, and through 180 days after the last dose of durvalumab. Should a
             woman become pregnant or suspect she is pregnant while she or her partner is
             participating in this study, she should inform her treating physician immediately.

          -  Patient is willing and able to comply with the protocol for the duration of the study
             including undergoing treatment and scheduled visits and examinations including follow
             up.

          -  Human immunodeficiency virus (HIV)-positive patients are eligible provided they have a
             negative viral load, CD4 count > 250, and are on a stable antiretroviral regimen.

          -  Ability to understand and the willingness to sign a written informed consent document.
             Patients with impaired decision-making capacity who have a close caregiver or legal
             guardian are also eligible with the consent of the caregiver/guardian.

        Exclusion Criteria:

          -  Patients who are not considered to be candidates for carboplatin + gemcitabine for
             first line therapy of their metastatic triple negative breast cancer are not eligible.

          -  Patients who have had chemotherapy, radiotherapy (to more than 30% of the bone
             marrow), or biologic therapy within 30 days (42 days for nitrosoureas or mitomycin C)
             prior to entering the study.

          -  Patients who have received prior immunotherapy for metastatic disease.

          -  Any previous treatment with a PD1 or PD-L1 inhibitor, including durvalumab (MEDI4736)
             or tremelimumab.

          -  Patients who have not recovered from grade >= 2 adverse events due to prior
             anti-cancer therapy with the exception of alopecia, vitiligo, and the laboratory
             values defined in the inclusion criteria.

               -  Patients with grade >= 2 neuropathy will be evaluated on a case-by-case basis
                  after consultation with the study physician.

          -  Patients who are receiving any other investigational agents or who have received an
             investigational agent within the last 30 days.

          -  Receipt of live attenuated vaccination within 6 months prior to study entry or within
             30 days of receiving durvalumab (MEDI4736) and tremelimumab.

               -  Note: Patients, if enrolled, should not receive live vaccine whilst receiving
                  study treatment and up to 30 days after the last dose of study treatment.

          -  Major surgical procedure within 28 days prior to the first dose of durvalumab
             (MEDI4736) and tremelimumab. Local surgery of isolated lesions for palliative intent
             is acceptable.

          -  Current or prior use of immunosuppressive medication within 14 days before the first
             dose of durvalumab (MEDI4736) or tremelimumab. The following are exceptions to this
             criterion:

               -  Intranasal, inhaled, topical steroids or local steroid injections (e.g.
                  intra-articular injection)

               -  Systemic corticosteroids at physiological doses which are not to exceed 10 mg/day
                  of prednisone or an equivalent corticosteroid

               -  Steroids as premedication for hypersensitivity reactions (e.g. CT scan
                  premedication).

          -  Known central nervous system (CNS) disease, except for treated asymptomatic CNS
             metastases. Patients with known brain metastases should be excluded from this clinical
             trial because of their poor prognosis and because they often develop progressive
             neurologic dysfunction that would confound the evaluation of neurologic and other
             adverse events.

          -  Leptomeningeal disease or history of leptomeningeal carcinomatosis.

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to durvalumab (MEDI4736) and tremelimumab. Known allergy, or history of
             serious adverse reaction to vaccines, such as anaphylaxis, hives or respiratory
             difficulty.

          -  Mean QT interval corrected for heart rate using Fridericia's formula (QTcF) >= 470 ms
             calculated from 3 electrocardiograms (ECGs) (within 15 minutes at 5 minutes apart).

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable
             angina pectoris, cardiac arrhythmia, interstitial lung disease, serious chronic
             gastrointestinal conditions associated with diarrhea, evidence of any acute or chronic
             viral illness or disease, or psychiatric illness/social situations that would limit
             compliance with study requirements.

          -  Pregnant women are excluded from this study because durvalumab (MEDI4736) and
             tremelimumab has 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 durvalumab (MEDI4736) and tremelimumab, breastfeeding
             should be discontinued if the mother is treated with durvalumab (MEDI4736) and
             tremelimumab. These potential risks may also apply to other agents used in this study.
             A negative serum pregnancy test is required no more than 7 days before study entry.

          -  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.

          -  History of pneumonitis or interstitial lung disease.

          -  History of active primary immunodeficiency.

          -  Active infection including tuberculosis (clinical evaluation that includes clinical
             history, physical examination, and radiographic findings, and TB testing in line with
             local practice), hepatitis B (known positive hepatitis B virus [HBV] surface antigen
             [HBsAg] result), or hepatitis C. Patients with a past or resolved HBV infection
             (defined as the presence of hepatitis B core antibody [anti-HBc] and absence of HBsAg)
             are eligible. Patients positive for hepatitis C (HCV) antibody are eligible only if
             polymerase chain reaction is negative for HCV ribonucleic acid (RNA).

          -  The patient with a previous history of non-breast malignancy is eligible for this
             study only if the patient meets the following criteria for a cancer survivor. A cancer
             survivor is eligible provided the following criteria are met:

               -  Patient has undergone potentially curative therapy for all prior malignancies.

               -  Patients have been considered disease free for at least 1 year (with the
                  exception of basal cell or squamous cell carcinoma of the skin or
                  carcinoma-in-situ of the cervix).

          -  Patients with a strong likelihood of non-adherence (such as difficulties in adhering
             to follow-up schedule due to geographic distance from the treatment facility) should
             not be knowingly registered.

          -  History of allogeneic organ transplantation.