Here are some products which support IMAP. More detailed listings can be found at The IMAP Connection
Pine: As far as I know the first text-based client to fully support IMAP. Pine uses the c-client code base which is used by most clients and servers to implement IMAP support.
Thunderbird: Thunderbird provides one of the best IMAP solutions of all the free e-mail clients out there. It supports disconnected mode, spam filtering, phishing protection, virtual folders, etc.
Evolution: Open source's answer to Outlook. Unfortunately they inherited some of Outlook's bad along with the good. That said, more recent versions seem to have all the right IMAP features supported and it seems to perform better than the 1.x version. I'll be testing this one out more in the future, as it seems like a real competitor to Thunderbird.
Microsoft Outlook Express: Outlook Express also supports IMAP. I've had two problems with it however. First of all, it's filtering support does not work on IMAP mail, and secondly it seems to want to open multiple connections to the IMAP mail server. Aside from wondering what the point of all this is, it causes problems for U of W IMAPd solutions, because the default configuration for this does not support multiple logins to the same mailbox.
Eudora: Eudora was one of the earliest supporters of IMAP from the PC/Mac market. Eudora also has support for IMSP/ACAP which I describe below.
Mulberry: One of the best Windows & Mac IMAP clients around. The only catch is that they are now out of business. :-(
Mutt: Mutt's IMAP support is still in the development stages. The stable version of Mutt barely supports IMAP. The development version of Mutt has more extensive support but still doesn't match what most of the other mail clients on this list provide. That being said, Mutt's macro facility can get around a lot of limitations. Of course, Mutt supports PGP as well, which is a nice touch.
Muttng A successor to mutt. Basically the developers working on Mutt decided it was getting too difficult to make the changes they wanted to make with the existing code base, so they started something new. Muttng's IMAP support is much better, particularly with the addition of an offline header cache.
ICEMail: ICEMail is written in Java, and takes advantage of Java Mail's built in support for IMAP. There are other Java IMAP clients, but this one is open source. It hasn't been updated in 3 years, so I'm not sure how much is going on there anymore.
UW IMAPd: This is the IMAPd that comes with most Linux distributions. If you installed it and you still don't have IMAP support, just uncomment the relevant line in /etc/services. This server is easy to get running, but on the downside by default still uses the old Unix mail format. This means you can't have multiple logins to the same account and you can't have sub folders. This version also reads your entire mail folder into memory, so it tends to be quite RAM hungry.
Cyrus IMAP Server: This is the server I currently use. The Cyrus Project at Carnagie-Mellon is exploring a lot new ideas in the area of remote mail, including ACAP (their implementation is written in Concurrent ML which gets them bonus points for truly geeking out) formerly known as IMSP. This basically move your MUA preferences on the server to! The Cyrus Server uses a message store that is a cross between a filesystem and a database, which in my experience produces fairly impressive performance results. Other cool things the Cyrus Server includes are ACL's on a per mailbox bases (including public mailboxes) SASL for secure authentication and Sieve for server based filtering of e-mail messages.
CommuniGate Pro: This sucker does it all. Probably one of the most feature rich IMAP/mail servers available for Linux, but unfortunately it's commercial, so you have to pay for it.
Lotus Notes, Microsoft Exchange, Netscape Messaging Server, and Sun Internet Mail Server: I mention these just in case your IS department tells you that they can't support IMAP. All of these packages support IMAP out of the box.