Inclusion Criteria:  - 1. Disease Related Criteria  1.1 Dose Escalation Phase  Subjects must have histologically proven bladder cancer of any histology and any stage or grade confirmed by biopsy or transurethral bladder tumour resection (TURBT) within 120 days of study registration (defined as date of signing informed consent). Subjects must be scheduled for radical cystectomy at the trial hospital.  1.2 Dose Expansion Phase  Subjects must have histologically proven, recurrent, urothelial carcinoma in situ of the bladder within 60 days prior to registration. Pure squamous carcinoma in situ will be excluded. Subjects may have concomitant Ta or T1 bladder tumours that have undergone visible complete resection within 60 days prior to registration. CIS disease is not expected to be completely excised. All patient tumour tissue from the histologic diagnosis of recurrence is available for central review by the study pathologist at Vancouver General Hospital. This does not need to be completed before starting study treatment. Subjects must have had cystoscopy confirming no visible papillary tumour within 21 days prior to registration. (CIS disease is not expected to have been completely excised). If the TURBT or bladder biopsy falls within 21 days of registration it will fulfil this criterion. Subjects must have had urine cytology within 21 days prior to registration. Cytology for subjects with CIS component is not expected to be negative for malignant cells. All subjects with T1 urothelial carcinoma at study entry must undergo re-TURBT within 60 days prior to registration, and must have evidence of uninvolved muscularis propria in the pathologic specimen from either the first or the second TURBT. Tissue from the re-resection must be submitted for central review in addition to the tissue from the first TURBT (see Section 8.2). The TURBT that identified the recurrent T1 disease may have taken place more than 60 days prior to registration but not more than 120 days. Subjects with high-grade Ta or CIS do not require a re-TURBT, but if this is performed at the discretion of the treating physician, the second TURBT must be within 60 days of registration. There is no requirement for muscularis propria in the specimen of Ta/CIS subjects, but the tissue from the first and second TURBTs must be submitted for central review. If a patient with Ta/T1 disease undergoes repeat TURBT, the patient can be included as having CIS if there is CIS on either TURBT. Subjects with prior urothelial carcinoma in the upper urinary tract within the previous 24 months will only be eligible if they had ≤ T1 carcinoma and were treated with nephroureterectomy. Subjects must have a CT or MRI (including CT-IVP, CT-urogram or MR-urogram) of the abdomen and pelvis to rule out upper tract malignancy and intra-abdominal metastases within 90 days prior to registration. If a patient cannot tolerate intravenous contrast, a retrograde pyelogram should be performed within 90 days prior to registration.  Subjects must be deemed unfit for radical cystectomy by the treating physician, or the patient must refuse radical cystectomy, which is considered standard of care for these subjects. The reason for subjects not to undergo cystectomy will be clearly documented.  2. Prior/Concurrent Therapy Criteria (Dose Expansion Phase only)  a. For the Dose Expansion Phase of the trial, subjects must be BCG-unresponsive. A patient is BCG-unresponsive if they meet one or both of the following criteria:  - Patient has persistent or recurrent CIS after completing therapy with at least induction BCG (≥ 5 doses) and first round maintenance (≥ 2 doses) or second induction BCG (≥ 2 doses). Both rounds of BCG must have been administered within a 12-month period. Trial registration must occur within 9 months of the last dose of BCG.  If a patient does not meet these criteria only because the last dose of BCG was more than 9 months ago, the patient may become eligible if he/she shows histologically proven CIS after an additional round of induction or maintenance BCG (≥ 3 doses) within 9 months prior to registration.  - Patient achieves disease-free state at 6-month time point (i.e., complete response; presence of only low-grade tumour at this timepoint is still considered complete response) after induction and maintenance (or second round of induction) BCG but later experiences CIS recurrence (with or without concomitant a high-grade Ta/T1) within 12 months after the last dose of BCG. The time of eligibility is measured from the last dose of BCG to the time of disease recurrence. The patient must be registered on the trial within 60 days of this recurrence, or within 60 days of a re-TURBT if indicated.  3. Clinical/Laboratory Criteria (For Dose Escalation and Expansion Phases)  Capable of giving signed informed consent which includes compliance with the requirements and restrictions listed in the informed consent form (ICF) and in this protocol. Written informed consent must be obtained from the patient/legal representative prior to performing any protocol-related procedures, including screening evaluations. Age >18 years at time of study entry Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 Body weight >30 kg  Adequate normal organ and marrow function as defined below (These results must be obtained within 42 days prior to registration):  - Haemoglobin ≥9.0 g/dL  - Absolute neutrophil count (ANC) ≥1.5 x 109/L (> 1500 per mm3)  - Platelet count ≥100 x 109/L (>75,000 per mm3)  - Serum bilirubin ≤1.5 x institutional upper limit of normal (ULN). This will not apply to subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology), who will be allowed only in consultation with their physician.  - AST (SGOT)/ALT (SGPT) ≤2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be ≤5x ULN  Measured creatinine clearance >40 mL/min or Calculated creatinine clearance >40 mL/min by the Cockcroft-Gault formula (Cockcroft and Gault 1976) or by 24-hour urine collection for determination of creatinine clearance:  Males:  Creatinine clearance (mL/min) = Weight (kg) x (140 - Age) 72 x serum creatinine (mg/dL)  Females:  Creatinine clearance (mL/min) = Weight (kg) x (140 - Age) x 0.85 72 x serum creatinine (mg/dL)  Evidence of post-menopausal status or negative urinary or serum pregnancy test for female pre-menopausal subjects. The intention of this restriction is to prevent or minimize embryo/fetal exposures to study drug. 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).  Subjects must have a baseline ECG performed within 42 days prior to registration. 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.  Must have a life expectancy of at least 1 year.  Exclusion Criteria:  - 1. Involvement in the planning and/or conduct of the study (applies to both AstraZeneca staff and/or staff at the study site) 2. Subjects must not have had urothelial carcinoma in the prostatic urethra within the previous 24 months.  3. Subjects must not have had muscle invasive (stage T2 or greater) or lymph node positive (N1-3) urothelial carcinoma of the bladder or upper tract (ureter, renal pelvis, renal calyx) at any time.  4. Participation in another clinical study with an investigational product during the last 28 days 5. Concurrent enrolment in another clinical study, unless it is an observational (non-interventional) clinical study or during the follow-up period of an interventional study 6. Subjects must not have had prior systemic chemotherapy for bladder cancer or systemic immunotherapy, including, but not limited to interferon alfa-2b, high dose IL-2, PEG-IFN, anti-PD-1, anti-PD-L1, or anti-CTLA4.  7. Receipt of the last dose of anticancer therapy (chemotherapy, immunotherapy, endocrine therapy, targeted therapy, biologic therapy, tumour embolization, monoclonal antibodies, radiotherapy) ≤28 days prior to the first dose of study drug. If sufficient wash-out time has not occurred due to the schedule or pharmacokinetic properties of an agent, a longer wash-out period will be required, as agreed by AstraZeneca/MedImmune and the Qualified Investigator (QI) 5. Any unresolved toxicity grade ≥2 (National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE)) from previous anticancer therapy (including surgery and intravesical therapy) with the exception of alopecia, vitiligo, and the laboratory values defined in the inclusion criteria  - Subjects with Grade ≥2 neuropathy will be evaluated on a case-by-case basis after consultation with the QI.  - Subjects with irreversible toxicity not reasonably expected to be exacerbated by treatment with durvalumab or tremelimumab may be included only after consultation with the QI.  6. Any concurrent chemotherapy, IP, biologic, or hormonal therapy for cancer treatment. Concurrent use of hormonal therapy for non-cancer-related conditions (e.g., hormone replacement therapy) is acceptable. Subjects must not require treatment with a RANKL inhibitor (e.g. denosumab) who cannot discontinue it before treatment with durvalumab/tremelimumab.  7. Subjects must not have received any prior radiation to the bladder for bladder cancer.  8. Major surgical procedure (as defined by the QI) within 28 days prior to the first dose of IP. Note: Local surgery including bladder biopsy or transurethral bladder tumour resection is acceptable.  9. History of allogenic bone marrow or solid organ transplantation. 10. Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease [e.g., colitis or Crohn's disease], diverticulitis [with the exception of diverticulosis], systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome [granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc]). The following are exceptions to this criterion:  - Subjects with vitiligo or alopecia  - Subjects with hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement  - Any chronic skin condition that does not require systemic therapy  - Subjects without active disease in the last 5 years may be included but only after consultation with the study physician  - Subjects with celiac disease controlled by diet alone 11. Uncontrolled intercurrent illness, including but not limited to, ongoing or active infection (requiring oral or IV antibiotics within 14 days prior to registration), symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea, or psychiatric illness/social situations that would limit compliance with study requirement, substantially increase risk of incurring AEs or compromise the ability of the patient to give written informed consent 12. History of another primary malignancy except for  - Malignancy treated with curative intent and with no known active disease ≥5 years before the first dose of IP and of low potential risk for recurrence  Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease Adequately treated carcinoma in situ without evidence of disease Low risk prostate cancer (according to National Comprehensive Cancer Network (NCCN) risk classification) on active surveillance 13. History of active primary immunodeficiency 14. Subjects positive for HIV are eligible only if they have all of the following: A CD4 count above 250 cells/mcL and an undetectable HIV viral load on standard PCR-based tests. A stable regimen of highly active anti-retroviral therapy (HAART)  No requirement for concurrent antibiotics or antifungal agents for the prevention of opportunistic infections 15. 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 HBV surface antigen (HBsAg) result), hepatitis C, Subjects with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody [anti-HBc] and absence of HBsAg) are eligible. Subjects positive for hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative for HCV RNA.  16. Current or prior use of immunosuppressive medication within 14 days before the first dose of durvalumab 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 physiologic doses not to exceed 10 mg/day of prednisone or its equivalent  Steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication) 17. Receipt of live attenuated vaccine within 30 days prior to the first dose of IP. Note: Subjects, if enrolled, should not receive live vaccine whilst receiving IP and up to 30 days after the last dose of IP.  18. Female subjects who are pregnant or breastfeeding or male or female subjects of reproductive potential who are not willing to employ effective birth control from screening to 90 days after the last dose of durvalumab monotherapy or 180 days after the last dose of durvalumab + tremelimumab combination therapy.  19. Known allergy or hypersensitivity to any of the study drugs or any of the study drug excipients.  20. Prior randomisation or treatment in a previous durvalumab and/or tremelimumab clinical study regardless of treatment arm assignment.  21. Judgment by the investigator that the patient is unsuitable to participate in the study and the patient is unlikely to comply with study procedures, restrictions and requirements.  22. Urethral stricture disease, other infravesical obstruction or other condition that will significantly impede cystoscopic injection of tremelimumab. This would include also oral anticoagulation other than 81 mg acetylsalicylic acid or non-steroidal anti-inflammatory drug (NSAID) if the patient is not able to interrupt anticoagulation or use low molecular weight heparin for bridging around the time of cystoscopic injection of tremelimumab