Primary Objectives * For Escalation Phase: Evaluate safety, tolerability and recommended phase 2 dose (RP2D) of DS-8201a in combination with AZD6738 in advanced solid tumors with HER2 expression. * For Expansion Phase: Assess differential pharmacodynamic (PD) profile of tumor tissue (DNA damage repair) between Top1 inhibition and dual inhibition of Top1 and ATR in patients with colorectal cancer and gastroesophageal cancer with HER2 expression. Secondary Objectives * To observe and record anti-tumor activity. Although the clinical benefit of this drug combination has not yet been established, the intent of offering this treatment is to provide a possible therapeutic benefit, and thus the patient will be carefully monitored for tumor response and symptom relief in addition to safety and tolerability. * To evaluate pharmacokinetics (PK) of DS-8201a and AZD6738 and immunogenicity of DS8201a. Exploratory Objectives * Evaluate association between HER2 heterogeneity and response to DS-8201a + AZD6738 therapy using central protein expression assessment. * Determine predictive biomarkers (including but not restricted to: HER2 protein levels, HER2 gene copy number, alterations of TP53, ATM and RAS) of DS-8201a + AZD6738 efficacy in advanced solid tumors (specifically colorectal cancer and gastroesophageal cancer) with HER2 expression. * To characterize PD biomarkers for efficacy of DS-8201a + AZD6738 efficacy in advanced solid tumors (specifically colorectal cancer and gastroesophageal cancer) with HER2 expression (such as pRAD50 and SLFN11). * To establish a biorepository of tissue, blood and pre-clinical models (PDXs) for HER2 expressing advanced solid tumors (specifically colorectal cancer and gastroesophageal cancer).