Short answer - de facto limit of 2000 characters
If you keep URLs under 2000 characters, they'll work in virtually any combination of client and server software.
If you are targeting particular browsers, see below for more details on specific limits.
Longer answer - first, the standards...
RFC 2616 (Hypertext Transfer Protocol HTTP/1.1) section 3.2.1 says
The HTTP protocol does not place any a priori limit on the length of a URI. Servers MUST be able to handle the URI of any resource they serve, and SHOULD be able to handle URIs of unbounded length if they provide GET-based forms that could generate such URIs. A server SHOULD return 414 (Request-URI Too Long) status if a URI is longer than the server can handle (see section 10.4.15).
That RFC has been obsoleted by RFC7230 which is a refresh of the HTTP/1.1 specification. It contains similar language, but also goes on to suggest this:
Various ad hoc limitations on request-line length are found in practice. It is RECOMMENDED that all HTTP senders and recipients support, at a minimum, request-line lengths of 8000 octets.
...and the reality
That's what the standards say. For the reality, there was an article on boutell.com (link goes to Internet Archive backup) that discussed what individual browser and server implementations will support. The executive summary is:
Extremely long URLs are usually a mistake. URLs over 2,000 characters will not work in the most popular web browsers. Don't use them if you intend your site to work for the majority of Internet users.
(Note: this is a quote from an article written in 2006, but in 2015 IE's declining usage means that longer URLs do work for the majority. However, IE still has the limitation...)
Internet Explorer's limitations...
I've tested IE10 and the address bar will only accept 2083 chars. You can click a URL which is longer than this, but the address bar will still only show 2083 characters of this link.
There's a nice writeup on the IE Internals blog which goes into some of the background to this.
There are mixed reports IE11 supports longer URLs - see comments below. Given some people report issues, the general advice still stands.
Search engines like URLs < 2048 chars...
Be aware that the sitemaps protocol, which allows a site to inform search engines about available pages, has a limit of 2048 characters in a URL. If you intend to use sitemaps, a limit has been decided for you! (see Calin-Andrei Burloiu's answer below)
There's also some research from 2010 into the maximum URL length that search engines will crawl and index. They found the limit was 2047 chars, which appears allied to the sitemap protocol spec. However, they also found the Google SERP tool wouldn't cope with URLs longer than 1855 chars.
CDNs have limits
CDNs also impose limits on URI length, and will return a
414 Too long request when these limits are reached, for example:
(credit to timrs2998 for providing that info in the comments)
Additional browser roundup
Browser Address bar document.location or anchor tag ------------------------------------------ Chrome 32779 >64k Android 8192 >64k Firefox >64k >64k Safari >64k >64k IE11 2047 5120 Edge 16 2047 10240
See also this answer from Matas Vaitkevicius below.
Is this information up to date?
This is a popular question, and as the original research is ~14 years old I'll try to keep it up to date: As of Jan 2021, the advice still stands. Even though IE11 may possibly accept longer URLs, the ubiquity of older IE installations plus the search engine limitations mean staying under 2000 chars is the best general policy.