Zimbra includes both a webmail client and an email server, so if you're looking for an all-in-one solution, it may be a good choice.

Zimbra is a well-maintained project that has been hosted at a number of different corporate entities through the years, and was acquired by Synacore in 2016. It features most of the things you've come to expect in a modern webmail client, from webmail to folders to contact lists to a number of pluggable extensions, and generally works very well. I have to admit that I'm most familiar with an older version of Zimbra, which felt at times slow and clunky, especially on mobile, but it appears that more recent versions have overcome these issues and provide a snappy, clean interface regardless of the device you are using. A desktop client is also available for those who prefer a more native experience.