Inclusion Criteria:

          1. Obtained written informed consent

          2. Male or female aged at least 18 years. Aged at least 20 if Japanese.

          3. Histologically or cytologically confirmed diagnosis of NSCLC with single activating
             EGFR mutations (L858R or Exon19Del).

          4. Eastern Cooperative Oncology Group performance status of 0 to1. For LM patients, 0 to
             2 is acceptable.

          5. In Part A, prior treatment with at least one line of a single agent EGFR TKI and at
             least 1 line of chemotherapy.

          6. In Part B-BM expansion, patients must have not received any EGFR TKI and have
             asymptomatic brain metastasis, either found during screening process which does not
             require local treatment in the opinion of the investigator or local treatment has been
             given (surgery or radiation), patient must be stable without corticosteroid and/or
             anti-convulsants treatment for at least 2 weeks before study enrollment. For Part B-LM
             expansion, patients who received previous EGFR TKI treatment must have stable
             extracranial disease;EGFR TKI treatment naïve patients can also be enrolled into
             AZD9291 cohorts, or AZD3759 cohorts if efficacy signal seen in Part A and agreed by
             Safety Review Committee.

          7. For patients with neither LM nor measurable BM: At least one measurable extracranial
             lesion. For patients with measurable BM but without LM: at least one measurable
             intracranial lesion

          8. For patients with LM: Confirmed diagnosis of LM by positive CSF cytology.

          9. Male patients should be willing to use barrier contraception, i.e., condoms, until 3
             months after last study drug is taken.

         10. Females should agree to use adequate contraceptive measures, should not be breast
             feeding and must have a negative pregnancy test prior to start of dosing if of
             child-bearing potential or must have evidence of non-child-bearing potential

         11. In Part B-AZD9291 LM expansion (sub-cohort of T790M+ LM patients), patients must have
             central confirmation of T790M+ mutation status from a sample taken after documented
             progression on the last treatment administered prior to enrolling in the study.
             Patients must have received prior therapy with an EGFR TKI and may also have received
             additional lines of treatment. Stable extracranial disease is not required.

        Exclusion Criteria:

          1. For patients with LM and/or BM, CNS complications that require urgent neurosurgical
             intervention

          2. For patient with LM, inability to undergo collection of CSF

          3. Treatment with an EGFR TKI (e.g., erlotinib or gefitinib) within 8 days or
             approximately 5 x half-life, whichever is the longer, of the first dose of study
             treatment.

          4. Any cytotoxic chemotherapy,or other anticancer drugs for the treatment of advanced
             NSCLC from a previous treatment regimen within 14 days of the first dose of study
             treatment

          5. Radiotherapy with a wide field of radiation within 4 weeks or radiotherapy with a
             limited field of radiation for palliation within 1 week of the first dose of study
             treatment with the exception of patients receiving radiation to more than 30% of the
             bone marrow which must be completed within 4 weeks of the first dose of study
             treatment.

          6. Patients currently receiving (or unable to stop use at least 1 week prior to receiving
             the first dose of AZD3759/AZD9291) medications or herbal supplements known to be
             potent inhibitors or inducers of cytochrome P450 3A4/5 and potential inhibitors of
             cytochrome P450 2C8 (for patients to be enrolled into AZD9291 cohorts only).

          7. Past medical history of interstitial lung disease, drug-induced interstitial lung
             disease, radiation pneumonitis which required steroid treatment, or any evidence of
             clinically active interstitial lung disease.

          8. Known intracranial haemorrhage which is unrelated to tumour

          9. Refractory nausea and vomiting if not controlled by supportive therapy, chronic
             gastrointestinal diseases, inability to swallow the formulated product or previous
             significant bowel resection that would preclude adequate absorption of AZD3759/AZD9291

         10. Any evidence of severe or uncontrolled systemic diseases, including uncontrolled
             hypertension and active bleeding diatheses

         11. Inadequate bone marrow reserve or organ function

         12. Clinically significant ECG abnormalities or any factors that increase the risk of
             corrected QT interval prolongation or risk of arrhythmic events

         13. Prior history of whole brain radiotherapy (only applicable for AZD3759 BM expansion)