IMAP account through GMail or Live?

I currently use an IMAP account running from my own domain as my primary e-mail. I came across some articles mentioning Gmail as being able to import POP3 accounts. Do Windows Live (Hotmail) or Gmail allow you to do this with an IMAP account?

Basically what I'm looking for is a way to use my IMAP account as webmail across any computer. My current host has a webmail option, but the UI is a disgrace.

If your host allows you to run code, you could consider installing your own webmail interface like Horde/IMP or SquirrelMail or something.

I don't think you'll find any of the existing e-mail services supporting IMAP client mode, it's just too difficult to integrate. POP is easy since it's a dumb mailbox that you just poll and throw any messages in the inbox. IMAP is constantly changing and you need to track state to do anything useful with it, which would be a UI nightmare and technically difficult to implement as part of an existing mail service. Maybe there are some web-based IMAP clients out there that are just IMAP clients and let you connect to your own mail servers, but that's rather a niche and I haven't heard of any.

If you want to use the GMail interface you're probably better off just forwarding your mail there and setting up a custom from address, or setting up Apps on your domain.