Inclusion Criteria:

          -  STEP 1 REGISTRATION AND RANDOMIZATION

          -  Patient must have the ability to understand and the willingness to sign a written
             informed consent document. Patients with impaired decision-making capacity (IDMC) who
             have a legally authorized representative (LAR) or caregiver and/or family member
             available will also be considered eligible

          -  Patient must have a diagnosis of high grade upper tract urothelial carcinoma proven by
             biopsy within 60 days prior to registration with one of the following:

               -  Upper urinary tract mass on cross-sectional imaging or

               -  Tumor directly visualized during upper urinary tract endoscopy before referral to
                  medical oncology

                    -  NOTE: Biopsy is standard of care (SOC) and required for enrollment to study.
                       This is vital for best practice

          -  Leukocytes >= 3,000/mcL (obtained =< 14 days prior to registration)

          -  Platelets >= 100,000/mcL (obtained =< 14 days prior to registration)

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (or =< 2.5 x ULN
             for patients with Gilbert's disease) (obtained =< 14 days prior to registration)

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional ULN (obtained =< 14 days prior to registration)

          -  Hemoglobin (Hgb) >= 9 g/dL (obtained =< 14 days prior to registration)

               -  NOTE: Packed red blood transfusion is allowed to achieve this parameter as per
                  treating investigator

          -  Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
             therapy with undetectable viral load within 6 months are eligible for this trial

               -  NOTE: These patients must be stable on their anti-retroviral regimen with
                  evidence of at least two undetectable viral loads within the past 6 months on the
                  same regimen; the most recent undetectable viral load must be within the past 12
                  weeks. They must have a CD4 count of greater than 250 cells/mcL over the past 6
                  months on this same anti-retroviral regimen and must not have had a CD4 count <
                  200 cells/mcL over the past 2 years, unless it was deemed related to the cancer
                  and/or chemotherapy induced bone marrow suppression. They must not be currently
                  receiving prophylactic therapy for an opportunistic infection and must not have
                  had an opportunistic infection within the past 6 months

               -  NOTE: For patients who have received chemotherapy in the past 6 months, a CD4
                  count < 250 cells/mcL during chemotherapy is permitted as long as viral loads
                  were undetectable during this same chemotherapy. They must have an undetectable
                  viral load and a CD4 count >= 250 cells/mcL within 7 days of registration

          -  For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral
             load must be undetectable on suppressive therapy, if indicated

               -  NOTE: Testing for HIV, hepatitis B or hepatitis C is not required unless
                  clinically indicated

          -  Patients with a history of hepatitis C virus (HCV) infection must have been treated
             and have undetectable viral load. For patients with HCV infection who are currently on
             treatment, they are eligible if they have an undetectable HCV viral load

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, patients should be class 2B or better

          -  Patient must have a body weight of > 30 kg

          -  Patient must have a life expectancy of >= 12 weeks

          -  Patient must have a creatinine clearance > 15 ml/min as by Crockroft-Gault or 24-hour
             creatinine clearance within 28 days prior to registration

               -  NOTE: Patients will be assigned to cisplatin-ineligible and cisplatin-eligible
                  cohorts based on their creatinine clearance, Eastern Cooperative Oncology Group
                  (ECOG) performance status, and grade (if any) of peripheral neuropathy and
                  hearing loss in keeping with SOC cisplatin contraindications. Patients that are
                  cisplatin-eligible will be randomized to either Arm A or Arm B

                    -  Patients that meet the following criteria will be assigned to the
                       cisplatin-ineligible Arm C:

                         -  Creatinine clearance of > 15 ml/min and =< 50 ml/min

                         -  Patient must have an absolute neutrophil count (ANC) >= 1,000/mcL
                            obtained =< 14 days prior to registration

                         -  Patient must have ECOG performance status 0-2

                    -  Patients that meet the following criteria will be randomized to
                       cisplatin-eligible Arm A or Arm B:

                         -  Patient must have an absolute neutrophil count (ANC) >= 1,500/mcL
                            obtained =< 14 days prior to randomization

                         -  Patient must have ECOG performance status 0-1

                         -  Patient must have left ventricular ejection fraction (LVEF) >= 50% by
                            (either multigated acquisition scan [MUGA] or 2-D echocardiogram)
                            obtained within 28 days prior to randomization

                         -  Patient must not have peripheral neuropathy >= grade 2 or hearing loss
                            >= grade 3

        Exclusion Criteria:

          -  Patients must not have any component of small cell carcinoma. Other variant histologic
             types are permitted provided the predominant (>= 50%) subtype is urothelial carcinoma

          -  Patients must not be pregnant or breast-feeding due to the potential harm to an unborn
             fetus and possible risk for adverse events in nursing infants with the treatment
             regimens being used. All patients of childbearing potential must have a blood test or
             urine study within 14 days prior to registration to rule out pregnancy. A patient of
             childbearing potential is defined as any patient, regardless of sexual orientation or
             whether they have undergone tubal ligation, who meets the following criteria: 1) has
             achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral
             oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer
             therapy does not rule out childbearing potential) for at least 24 consecutive months
             (i.e., has had menses at any time in the preceding 24 consecutive months)

          -  Patients of childbearing potential and sexually active patients must not expect to
             conceive or father children by using accepted and effective method(s) of contraception
             or by abstaining from sexual intercourse from the time of registration, while on study
             treatment and for at least 6 months after the last dose of protocol treatment

          -  Patient must have no evidence of metastatic disease or clinically enlarged lymph nodes
             (>= 1.0 cm short axis) on imaging required within 28 days prior to registration
             (solitary slightly enlarged lymph node with negative biopsy is allowed)

               -  NOTE: Patients with elevated alkaline phosphatase, calcium or suspicious bone
                  pain/tenderness should also undergo baseline bone scans to evaluate for bone
                  metastasis

          -  Patient must not have another active (or within 2 years) second malignancy other than
             resected non-melanoma skin cancers, resected in situ breast, cervical or other in situ
             carcinoma, and either clinically insignificant per the investigator (e.g. =< Gleason
             3+4) on surveillance or previously treated prostate cancer with no rising prostate
             specific antigen (PSA) and no plan to treat

               -  NOTE: Patients with a prior or concurrent malignancy whose natural history or
                  treatment does not have the potential to interfere with the safety or efficacy
                  assessment of the investigational regimen are eligible for this trial. Patients
                  in whom concomitant or prior bladder/urethra predominant (>= 50%) urothelial
                  carcinoma have been surgically resected and demonstrated to be only non-invasive
                  cancer (< cT1N0) are eligible regardless of time elapsed

          -  Patient must not have any uncontrolled illness including, but not limited to, ongoing
             or active infection including tuberculosis (clinical evaluation that includes clinical
             history, physical examination and radiographic findings, and tuberculosis [TB] testing
             in line with local practice), symptomatic congestive heart failure (CHF), myocardial
             infarction (MI) in last 3 months, or unstable angina pectoris, significant
             uncontrolled cardiac arrhythmia, liver cirrhosis, interstitial lung disease, or
             psychiatric illness/social situations that would limit compliance with study
             requirements

          -  Patient must not have received prior radiation therapy to >= 25% of the bone marrow
             for other diseases

          -  Patient must not have received prior systemic anthracycline therapy

               -  NOTE: Patients who have received prior intravesical chemotherapy at any time for
                  non-muscle invasive urothelial carcinoma of the bladder are eligible

          -  Patient must not have an active autoimmune disease requiring immunosuppressive therapy
             within 2 years prior to registration or a history of inflammatory bowel disease
             (inflammatory bowel disease [IBD], colitis, or Crohn's disease), systemic lupus
             erythematosus, Sarcoidosis syndrome, Wegener syndrome or immune-related pneumonitis or
             interstitial lung disease. Patients with well-controlled hyper/hypothyroidism, celiac
             controlled by diet alone, diverticulosis, diabetes mellitus type I, vitiligo,
             alopecia, psoriasis, eczema, lichen planus, or similar skin/mucosa condition are
             eligible

          -  Patient must not be on or have used immunosuppressive medication within 14 days prior
             to the first dose of MEDI4736 (MEDI4736 (durvalumab). The following are exceptions to
             this criterion:

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

               -  Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of
                  prednisone or its equivalent at the time of enrollment

               -  Steroids as premedications for hypersensitivity reactions (e.g. computed
                  tomography [CT] scan premedication)

          -  Patient must not have a concomitant primary urothelial carcinoma of the bladder and/or
             urethra

               -  NOTE: Patients in whom concomitant or prior bladder/urethra predominant (>= 50%)
                  urothelial carcinoma have been surgically resected and demonstrated to be only
                  non-invasive cancer (< cT1N0) are eligible regardless of time elapsed

          -  Patient must not have prior history of muscle-invasive urothelial carcinoma with or
             without systemic chemotherapy (T2-4a and/or N1) within 2 years prior to registration

               -  NOTE: Patients who have no evidence of disease (NED) for more than 2 years from
                  the latest therapy (surgery, radiation, chemotherapy, or clinical trial) are
                  eligible

          -  Patient must not have received live attenuated vaccine within 30 days prior to the
             first dose of MEDI4736 (durvalumab), while on protocol treatment and within 30 days
             after the last dose of MEDI4736 (durvalumab)

          -  Patient must not have had a major surgical procedure (as defined by the Investigator)
             within 28 days prior to registration

          -  Patient must not have a history of allogenic organ transplantation