Roll Over StudY for Patients Who Have Completed a Previous Oncology Study With Osimertinib (TAGRISSO) (ROSY-T)

Inclusion Criteria: 1. Provision of signed and dated, written Informed Consent Form (ICF). 2. Patient is currently deriving clinical benefit from continued treatment with osimertinib in an AstraZeneca parent study using osimertinib monotherapy which has met its endpoints or has otherwise stopped. 3. Patients should be using adequate contraceptive measures. Exclusion Criteria: 1. Ongoing, unresolved, Grade 3 or above toxicity requiring interruption of treatment at the time of the termination of the parent study. 2. Currently receiving treatment with any prohibited medication(s). 3. Concurrently enrolled in any other type of medical research judged not to be scientifically, or medically compatible with this study. 4. Permanent discontinuation from the parent study due to toxicity or disease progression. 5. Local access to commercially-available drug at no cost to the patient is permitted by local regulation. Exclusion Criteria for the sub-study: 1\. Active infection including active hepatitis C and Human immunodeficiency virus (HIV) infection or active uncontrolled Hepatitis B virus (HBV) infection. Screening for chronic conditions is not required. Patients with HBV infection are only eligible if they meet all the following criteria: * Demonstrated absence of HCV co-infection or history of HCV co-infection; * Demonstrated absence of HIV infection; * Patients receiving anti-viral treatment for at least 6 weeks prior to study treatment, HBV DNA is suppressed to \< 100 IU/mL, and transaminase levels are below ULN. Patients with a resolved or chronic HBV infection are eligible if they are: * Negative for HBsAg and positive for anti-HBc IgG. In addition, patients must be receiving anti-viral prophylaxis for 2 to 4 weeks prior to study treatment and 6 to 12 months (to be determined by hepatologist) post treatment; or * Positive for HBsAg, but for \> 6 months have had transaminases levels below ULN and HBV DNA levels below \< 100 IU/mL (ie, patients are in an inactive carrier state). In addition, patients must be receiving anti-viral prophylaxis for 2 to 4 weeks prior to study treatment. Should participants with HIV infection be included, patients are only eligible if they meet all the following criteria: * Undetectable viral RNA load for 6 months * CD4+ count of \> 350 cells/μL * No history of AIDS-defining opportunistic infection within the past 12 months (to be determined by hepatologist) post treatment * Stable for at least 4 weeks on anti-HIV medications.