Inclusion Criteria:

          1. Adults, aged 18 years and older with histologically and/or cytologically confirmed
             non-squamous non-small cell lung cancer with EGFR mutation of Exon 19 deletion or Exon
             21 L858R point mutation.

               -  Patients with co-mutations are allowed as long as one of the mutation is either
                  Exon 19 deletion or Exon 21 L858R point mutation

               -  Patients with mixed histology must have non-squamous NSCLC as the predominant
                  histology.

          2. Disease that has progressed and either:

             (i) No evidence of EGFR T790M resistance mutation in both tissue re-biopsy and plasma
             after one-line of EGFR tyrosine kinase inhibitor therapy (TKI) OR (ii) T790M mutation
             (detected in tissue re-biopsy, plasma or both) and progression on 3rd generation EGFR
             TKI; patients are allowed to have one prior line of TKI therapy before 3rd generation
             TKI

          3. Measurable disease according to RECIST 1.1

          4. ECOG performance status of 0 or 1

          5. Adequate bone marrow function as defined below (within 14 days prior to registration
             and with values within the ranges specified below):

               -  Platelets equal to or greater than 100 x 109/L

               -  Absolute neutrophil count (ANC) equal to or greater than 1.0 x 109/L (equal to or
                  greater than 1000 per mm3)

               -  Haemoglobin equal to or greater than 90 x g/L

          6. Adequate liver function (within 14 days prior to registration and with values within
             the ranges specified below):

               -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) equal to or
                  less than 2.5 x institutional upper limit of normal (ULN). (or equal to or less
                  than 5 x ULN if liver metastases are present)

               -  Bilirubin equal to or less than 1.5 x ULN

          7. Adequate renal function (within 14 days prior to registration):

             • Measured creatinine clearance greater than 40 mL/min or Calculated creatinine
             clearance greater than 40 mL/min by the Cockcroft-Gault formula or by 24-hour urine
             collection for determination of creatinine clearance

          8. Provision of a pre-treatment tumour tissue sample from a biopsy taken within 42 days
             prior to enrolment (core biopsy preferred) to determine NSCLC histology and for
             translational research.

          9. Evidence of post-menopausal status or negative urinary or serum pregnancy test for
             female pre-menopausal patients.

         10. 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.

         11. Signed written informed consent (main study and optional biobanking).

        Exclusion Criteria:

          1. Prior chemotherapy or immunotherapy, including prior anti-PD-1/anti-PD-L1 or
             anti-CTLA-4 antibodies, for advanced NSCLC.

               -  For recurrent, incurable disease, prior adjuvant chemotherapy or concurrent
                  chemotherapy and radiotherapy with curative intent is allowed, but must have been
                  completed more than 6 months ago, and must not have included treatment with an
                  immune checkpoint inhibitor.

               -  Prior EGFR-TKI (e.g. erlotinib, gefitinib, afatinib or osimertinib), including
                  experimental TKI agents, within 8 days or approximately 5 x half-life, whichever
                  is the longer, of the first dose of study treatment is allowed. (If sufficient
                  washout time has not occurred due to schedule or PK properties, an alternative
                  appropriate washout time based on known duration and time to reversibility of
                  drug related adverse events could be agreed upon by contacting the ILLUMINATE
                  Study Team).

          2. Mixed histology with any small cell or squamous component.

          3. Life expectancy of less than 3 months.

          4. Current enrolment or participation in another clinical study with an investigational
             product during the last 12 months, unless it is an observational (non-interventional)
             clinical study or during the follow-up period of an interventional study must first be
             discussed with ILLUMINATE Study Team before study enrolment.

          5. Any unresolved toxicity NCI CTCAE equal to or greater than Grade 2 from previous
             anticancer therapy with the exception of alopecia, vitiligo, and the laboratory values
             defined in the inclusion criteria.

               -  Patients with equal to or greater than Grade 2 neuropathy will be evaluated on a
                  case-by-case basis after consultation with the ILLUMINATE Study Team.

               -  Patients with irreversible toxicity not reasonably expected to be exacerbated by
                  treatment with durvalumab or tremelimumab may be included only after consultation
                  with the ILLUMINATE Study Team.

          6. Radiotherapy or major surgery (as defined by the local investigator) within 4 weeks of
             the first dose of study drug. Note: Local surgery on isolated lesions for palliative
             intent is acceptable.

          7. History of pneumonitis or pulmonary fibrosis with clinically significant impairment of
             pulmonary function.

          8. History of active primary immunodeficiency or allogeneic transplant.

          9. 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:

               -  Patients with vitiligo or alopecia

               -  Patients with hypothyroidism (e.g., following Hashimoto syndrome) stable on
                  hormone replacement

               -  Any chronic skin condition that does not require systemic therapy

               -  Patients without active disease in the last 5 years may be included but only
                  after consultation with the study physician

               -  Patients with celiac disease controlled by diet alone

         10. Concurrent illness, including severe infection that may jeopardise the ability of the
             patient to undergo the procedures outlined in this protocol with reasonable safety.

         11. History of another primary malignancy except for

               -  Malignancy treated with curative intent and with no known active disease equal to
                  or greater than 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 insitu without evidence of disease.

         12. Any history of leptomeningeal carcinomatosis, or untreated central nervous system
             metastases obtained during the screening period or identified prior to signing the
             PICF. Patients whose brain metastases have been treated may participate provided they
             show radiographic stability (defined as disease stability on imaging 4 weeks after
             commencing radiotherapy). Any neurologic symptoms that developed either as a result of
             the brain metastases or their treatment must have resolved or be stable either,
             without the use of steroids, or are stable on a steroid dose of equal to or less than
             10mg/day of prednisone or its equivalent (and anti-epileptic drugs) for at least 14
             days prior to the start of treatment.

         13. 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, or human immunodeficiency virus (positive HIV 1/2 antibodies). Patients
             with a past or resolved HBV infection (defined as the presence of hepatitis B core
             antibody [anti-HBc] and absence of HBsAg) are eligible. Patients positive for
             hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative
             for HCV RNA.

         14. 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)

         15. Live attenuated vaccine within 30 days prior to the first dose of study drug, whilst
             receiving study drug and up to 30 days after the last dose of study drug.

         16. Serious medical or psychiatric conditions that might limit the ability of the patient
             to comply with the protocol.

         17. Female patients who are pregnant or breastfeeding or male or female patients 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 plus tremelimumab combination therapy.

         18. Known allergy or hypersensitivity to any of the study drugs or excipients.