mail2forum does this very well, but it's only available for phpBB2. I've been using it successfully on a private, paid-membership forum with 1200 members for about 7 years now. It greatly increases participation, and like previous posters mentioned, advertising is not the only business model.
It works like this: when a post is made in a forum, each subscribed user gets it by email. Users can also reply to these emailed posts and they are then threaded into the discussion just as if they were made through the web interface. New topics can be started by email as well - users simply send a message to the forum's address. Each forum has a unique email address and users can choose from which forums they want to receive emails. mail2forum even handles attachments and can downsize big images so that users' mailboxes aren't clogged with big files.
Regex filtering does a good job of keeping out out-of-office replies and the like. To keep things clean and prevent replicating quotes from previous emails, "Reply above this line" can be used and the Regex filter would strip everything below it.
This feature combines the best of both worlds allowing members who prefer email communication to stay active and contribute, while those who prefer the web interface can participate online only.
Google Groups works like this, but they require a Gmail account for all participants.
I'm looking forward to seeing this feature implemented in a modern forum, not necessarily as a part of the core, but as a supported add-on.