How to stop Google from storing Google chat history

If you don't want Google to store your chat history and the Google chat history of the person you are chatting with, you can take the chat off the record and this is how you do that.

In the chat window you are having a conversation, click the More button (see picture) ...

... and choose Go off the record.

This means that from now on Google chats will not be stored in you Google chat history or in the Google chat history of the person you are chatting with.

And when you decide you don't mind Google storing chats you are having with that person any more, simply click More again and choose the Stop chatting off the record option.