Key Inclusion Criteria:

          -  Patient must be willing and capable of giving written Informed Consent, adhering to
             dosing and visit schedules, and meeting all study requirements

          -  Patient has histologically or cytologically confirmed locally advanced or metastatic
             non-small cell lung cancer (NSCLC) that is not amenable to treatment with curative
             intent

          -  Prior treatment status:

               -  Cohorts 1 and 2: Patient has had at least one prior systemic treatment for
                  locally advanced or metastatic NSCLC

               -  Cohorts 3 and 4: Patient is treatment-naïve for locally advanced or metastatic
                  NSCLC and eligible to receive first-line treatment with poziotinib as determined
                  by the Investigator. Adjuvant/neo-adjuvant therapies (chemotherapy, radiotherapy,
                  or investigational agents) are permissible as long as they end at least 15 days
                  prior to study entry.

               -  Cohort 5: Patients who meet the criteria for enrollment in Cohorts 1 to 4, but
                  the enrollment in the respective cohort has been closed

               -  Cohort 6: Patients with EGFR mutation-positive NSCLC who progressed while on
                  treatment with first-line osimertinib

               -  Cohort 7: Patient has had at least one prior systemic treatment for locally
                  advanced or metastatic NSCLC

          -  Specific mutations:

               -  Cohort 1 and 3: Documented EGFR exon 20 insertion mutation

               -  Cohort 2 and 4: Documented HER2 exon 20 insertion mutation

               -  Cohort 5: Documented EGFR or HER2 exon 20 insertion mutations

               -  Cohort 6: Documented acquired EGFR mutation (tested after osimertinib
                  progression)

               -  Cohort 7: Documented EGFR or HER2 activating mutations

          -  Patient has adequate organ function at Baseline

        Key Exclusion Criteria:

          -  Patient has had previous treatment with poziotinib or any other EGFR or HER2 exon 20
             insertion mutation-selective tyrosine kinase inhibitor (TKI) prior to study
             participation. The currently approved TKIs (ie, erlotinib, gefitinib, afatinib,
             osimertinib) are not considered to be exon 20 insertion-selective and are permissible
             (Cohorts 1 and 2).

          -  Patient is concurrently receiving chemotherapy, biologics, immunotherapy for cancer
             treatment; systemic anti-cancer treatment or investigational treatment should not be
             used within 2 weeks or 5 half lives, whichever is longer; local radiation therapy for
             bone pain may be allowed

          -  Patient has had other malignancies within the past 3 years, except for stable
             non-melanoma skin cancer, fully-treated and stable, early-stage prostate cancer, or
             carcinoma in situ of the cervix or breast without need of treatment

          -  Patient is pregnant or breast-feeding