How to develop and test an app that sends emails (without filling someone's mailbox with test data)?

I have a lot of apps that send email. Sometimes it's one or two messages at a time. Sometimes it's thousands of messages.

In development, I usually test by substituting my own address for any recipient addresses. I'm sure that's what everybody else does, until they get fed up with it and find a better solution.

I was thinking about creating a dummy SMTP server that just catches the messages and dumps them in a SQLLite database, or an mbox file, or whatever.

But surely such a tool already exists? How do you test sending email?