A Phase II Study of Durvalumab Treatment - Substudy A: In Patients Who Discontinued Prior Checkpoint Therapy Due to Immune Related Toxicity - Substudy B: For Continued Treatment (+/- Tremelimumab) of Patients Previously Enrolled to Completed CCTG Studies

Substudy A: Inclusion Criteria: * Patients must have histologically and/or cytologically confirmed solid tumour, that is advanced/ metastatic/recurrent or unresectable and for which no curative therapy exists. * Patients must live within Canada and have received durvalumab alone, or durvalumab in combination with tremelimumab, with or without chemotherapy/targeted therapy. Patients who have received other anti PD-1/PD-L1 agents +/- anti CTLA agents are eligible, providing full details of prior therapy, toxicity and management are available. Consult with CCTG for further details. * Patients must have previously discontinued immunotherapy due to an irAE. * Immune-related adverse events must have resolved to ≤ grade 1 or baseline and patient must have completed corticosteroid therapy at least 28 days prior to registration in this current study. * Complete response, partial response or prolonged stable disease (SD ≥ 8 weeks) to initial immunotherapy. Patients that received prior adjuvant/neoadjuvant/consolidation immunotherapy are eligible providing there has been at least a 6 month treatment free interval prior to enrollment and patient has received at least one standard-of-care chemotherapy regimen in the palliative setting (discuss with CCTG if chemotherapy is not considered standard of care or not indicated or patient refused/not eligible as such patients are eligible). * Patients must have a life expectancy of at least 12 weeks. * Tumour material may have already been submitted to CCTG for the initial trial. If an additional formalin fixed paraffin embedded tissue block (from their primary or metastatic tumour) is available from tissue collected after immunotherapy discontinuation, patients must have provided informed consent for the release of the block. All patients must have provided informed consent for correlative studies. If patients from non-CCTG trials or commercial use are eventually enrolled, tumour material is also required if available, preferably from tissue collected after immunotherapy discontinuation. * Presence of clinically and/or radiologically documented disease. All radiology studies must be performed within 28 days prior to enrollment (within 35 days if negative). Patients ideally should have measurable disease. * ECOG performance status 0 or 1 * Previous Therapy * Patients who received other relevant standard cancer therapies since discontinuing immunotherapy remain eligible for inclusion as follows: * Patients may have received prior cytotoxic chemotherapy following discontinuation of immunotherapy for irAE. * Patients may have received other prior therapies such as tyrosine kinase inhibitors or other targeted agents, following discontinuation of immunotherapy for irAE. * Patients may not have received subsequent immune checkpoint inhibitors (anti-PD-(L)1 and anti-CTLA-4) following discontinuation of immunotherapy for irAE. Vaccines and oncolytic viruses are permitted. * Patients must have recovered from all reversible toxicity related to prior chemotherapy or systemic therapy (unless grade 1, irreversible, or considered by investigator as not clinically significant) and have adequate washout as follows: Longest of one of the following: * Two weeks; * 5 half-lives for investigational agents; * Standard cycle length of standard therapies. * Prior external beam radiation is permitted provided a minimum of 28 days (4 weeks) have elapsed between the last dose of radiation and date of enrollment. Exceptions may be made for low-dose, non-myelosuppressive radiotherapy after consultation with CCTG. Concurrent radiotherapy is not permitted. * Previous surgery is permitted provided that a minimum of 28 days (4 weeks) have elapsed between any major surgery and date of enrollment, and that wound healing has occurred. * Absolute neutrophils ≥ 1.5 x 10\^9/L * Platelets ≥ 100 x 10\^9/L * Hemoglobin ≥ 90 g/L * Bilirubin ≤ 1.5 x ULN (upper limit of normal) * AST and ALT ≤ 2.5 x ULN - ≤ 5.0 x ULN (if patient has liver metastases) * Serum creatinine \< 1.25 x ULN or * Creatinine clearance ≥ 40 mL/min * Patient consent must be appropriately obtained in accordance with applicable local and regulatory requirements. Each patient must sign a consent form prior to enrollment in the trial to document their willingness to participate. * Patients must be accessible for treatment and follow up. Patients registered on this trial must be treated and followed at the participating centre. This implies there must be reasonable geographical limits (for example: 1 ½ hour's driving distance) placed on patients being considered for this trial. * In accordance with CCTG policy, protocol treatment is to begin within 2 working days of patient enrollment * Women/men of childbearing potential must have agreed to use a highly effective contraceptive method * Subjects should not donate blood while participating in this study, or for at least 90 days following the last infusion of durvalumab Exclusion Criteria: * In general, patients with prior grade 4 non-hematological, non-endocrine immune-related adverse events are not eligible. * History of primary immunodeficiency, history of allogenic organ transplant that requires therapeutic immunosuppression and the use of immunosuppressive agents within 28 days of enrollment. * Live attenuated vaccination administered within 30 days prior to enrollment or within 30 days of receiving durvalumab. * History of hypersensitivity to durvalumab or any excipient. * Any immune-related adverse event that required biologic agents such as infliximab, or mycophenolate motefil to manage. * Documented progressive disease (PD) while on initial immunotherapy. Exception: patients who had iUPD but continued on immunotherapy, and did not have documented iCPD within 8 weeks of discontinuing immunotherapy * Patients who have experienced untreated and/or uncontrolled cardiovascular conditions and/or have symptomatic cardiac dysfunction (unstable angina, congestive heart failure, myocardial infarction within the previous year or cardiac ventricular arrhythmias requiring medication, history of 2nd or 3rd degree atrioventricular conduction defects). Patients with a significant cardiac history, even if controlled, should have a LVEF ≥ 50%. * Concurrent treatment with other investigational drugs or anti-cancer therapy. * Patients with serious illnesses or medical conditions which would not permit the patient to be managed according to the protocol (including corticosteroid administration), or would put the patient at risk. This includes but is not limited to: * History of significant neurologic or psychiatric disorder which would impair the ability to obtain consent or limit compliance with study requirements. * Active infection requiring systemic therapy; (including any patient known to have active hepatitis B, hepatitis C or human immunodeficiency virus (HIV) or tuberculosis or any infection requiring systemic therapy). * Active peptic ulcer disease or gastritis. * Untreated symptomatic brain metastases or brain metastases in whom radiation or surgery is indicated. * Patients with diabetes mellitus are eligible but must be clinically stable on therapy (if applicable) and investigator and patient should be aware of the potential risk of immune mediated pancreatic toxicity and B cell destruction. * Pregnant or lactating women Substudy B: Inclusion Criteria * Patients must be currently enrolled and receiving active treatment on a treatment arm containing durvalumab +/- tremelimumab with or without maintenance pemetrexed with no contraindications to continue receiving their current study regimen according to the protocol to which the patient is currently enrolled. * For BR.34 ONLY: patients who have disease progression (iUPD) on durvalumab may receive one dose of tremelimumab (75 mg) along with their next durvalumab infusion as long as all the following criteria are met: * Patient is clinically stable * According to the judgement of the treating physician, the patient had clinical benefit while receiving tremelimumab in the induction phase on BR.34 * ECOG performance status of 0 or 1 * Laboratory values meet the criteria below: Absolute neutrophils ≥ 1.5 x 10\^9/L Platelets ≥ 100 x 10\^9/L Hemoglobin ≥ 90 g/L Bilirubin ≤ 1.5 x ULN (upper limit of normal)\* AST and ALT ≤ 2.5 x ULN (if liver metastases are present, ≤ 5.0 x ULN) Serum creatinine \< 1.25 x ULN or Creatinine clearance ≥ 45 mL/min