This study is being done to determine the highest dose of AZD6738 that can safely be taken in combination with DS-8201a and what effect will this combination have on DNA repair mechanisms in advanced colorectal or gastroesophageal cancer.

There are two parts in this study, a dose escalation part, and a dose expansion part. The study doctor will tell each participant what part they are in.

In the dose escalation part of the study, participants will receive 3-week cycles of DS-8201 and AZD6738(ceralasertib). The combination of AZD6738 (ceralasertib) and DS-8201a is not approved by the U.S. Food and Drug Administration (FDA) for the treatment of colorectal or gastroesophageal cancer, however DS-8201a is approved by the FDA as treatment of breast or gastric cancer that expresses a gene or protein called HER2. DS-8201a is given intravenously while AZD6738 (ceralasertib) is taken orally.

In the dose expansion part of the study, participants will either receive a combination of DS-8201a and AZD6738 (ceralasertib) every 3-weeks, or DS-8201a alone for one 3-week cycle and then a combination of DS-8201a and AZD6738 (ceralasertib) after that.

In both parts of the study, DS-8201a is given intravenously while AZD6738 (ceralasertib) is taken orally.

Participants will receive treatment until their cancer grows, they develop unmanageable side-effects, or they withdraw from the study.