Copilot Chat saves your conversations and lists them under Chats on the left, much as Claude does. From the menu next to a chat's title you can rename it ("Q3 client renewal", not "New chat 14") or delete it. Because the list keeps a running record, you don't need to leave a thread open just in case, and you can let a chat that went nowhere go.
Each surface keeps its own history: a chat you start in Copilot in Word doesn't show up in standalone Copilot Chat, and vice versa. Copilot reads the whole conversation every time you send a message, so start a new chat for a new topic; older, unrelated messages dilute the context. Copilot can also draw selectively on past chats, so delete a conversation you don't want it to refer back to.