• Part A: Safety and tolerability of DZD9008. [ Time Frame: 28 days after the first multiple dose ]
  To investigate the safety and tolerability of DZD9008 when given orally to patients with advanced NSCLC with EGFR or HER2 mutations; To establish Maximum Tolerated Dose (MTD) (if possible) and Recommended Phase 2 Dose (PR2D) of DZD9008 when given orally in patients with advanced NSCLC with EGFR or HER2 mutations.
• Part B: Objective Response Rate (ORR) according to RECIST 1.1. [ Time Frame: through the study completion, an average of around 1 year ]
  To evaluate anti-tumor activity of DZD9008 in advanced NSCLC patients with EGFR Exon20 insertion, HER2 Exon20 insertion or EGFR uncommon mutations at defined dose(s) (Part B)

Inclusion Criteria:

1. Aged at least 18 years old (≥ 20 if in Japan), be able to provide a signed and dated, written informed consent.
2. With documented histological or cytological confirmed locally advanced or metastatic NSCLC with EGFR or HER2 mutations.
3. (ECOG) performance status 0-1.
4. Predicted life expectancy ≥ 12 weeks
5. Patient must have measurable disease according to RECIST 1.1.
6. Patients with brain metastasis (BM) can be enrolled under the condition that BM is stable, neurologically asymptomatic and does not require corticosteroid treatment.
7. Adequate organ system function.

Exclusion Criteria:

1. For part B: Patients who have received prior treatment with Poziotinib or TAK788 or other EGFR/HER2 exon20 insertion inhibitors should be excluded. Prior treatment with currently approved EGFR TKIs for sensitizing or T790M resistance mutations, such as gefitinib, erlotinib, osimertinib, afatinib and dacomitinb, are allowed.
2. Treatment with EGFR or HER2 antibodies, major surgery (excluding placement of vascular access), or onco-immunotherapy (e.g. immune checkpoint inhibitors PD-1, PD-L1, CTLA-4) within 4 weeks before screening.
3. Any cytotoxic chemotherapy, investigational agents or other anticancer drugs from a previous treatment regimen or clinical study within 14 days before screening.
4. Radiotherapy with a limited field of radiation for palliation within 1 week of the first dose or with a wide field of radiation which must be completed within 4 weeks before screening.
5. Receiving (or unable to stop using) medications or herbal supplements known to be potent inhibitors or inducers of CYP3A within 2-3 weeks before screening.
6. Grapefruit, grapefruit juice, and orange marmalade (made with Seville oranges) within 1 week before screening.
7. Any unresolved toxicities from prior therapy greater than CTCAE grade 1 at the time of starting DZD9008 with the exception of alopecia and grade 2 prior platinum-therapy related neuropathy.
8. Spinal cord compression or leptomeningeal metastasis.
9. As judged by the investigator, any evidence of severe or uncontrolled systemic diseases, which would jeopardize compliance with the protocol, or active infection including hepatitis B, hepatitis C and human immunodeficiency virus (HIV).
10. Any of the following cardiac criteria: (1) Mean resting corrected QT interval (QTcF) > 470 msec obtained from 3 electrocardiograms (ECGs); (2) Any clinically significant abnormalities in rhythm, conduction or morphology of resting ECG, e.g., complete left bundle branch block, third degree heart block, and second-degree heart block, PR interval > 250 msec. (3) Any factors that increase the risk of QTcF prolongation, such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives or any concomitant medication known to prolong the QT interval
11. 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
12. Refractory nausea and vomiting, chronic gastrointestinal diseases, inability to swallow the formulated product or previous significant bowel resection that would preclude adequate absorption of DZD9008
13. History of hypersensitivity to active or inactive excipients of DZD9008 or drugs with a similar chemical structure or class to DZD9008
14. Women who are pregnant or breast feeding
15. Involvement in the planning and conduct of the study.
16. Judgment by the investigator that the patient should not participate in the study if the patient is unlikely to comply with study procedures, restrictions and requirements.