I wrote the UUID generator/parser for Ruby, so I consider myself to be reasonably well-informed on the subject. There are four major UUID versions:
Version 4 UUIDs are essentially just 16 bytes of randomness pulled from a cryptographically secure random number generator, with some bit-twiddling to identify the UUID version and variant. These are extremely unlikely to collide, but it could happen if a PRNG is used or if you just happen to have really, really, really, really, really bad luck.
Version 5 and Version 3 UUIDs use the SHA1 and MD5 hash functions respectively, to combine a namespace with a piece of already unique data to generate a UUID. This will, for example, allow you to produce a UUID from a URL. Collisions here are only possible if the underlying hash function also has a collision.
Version 1 UUIDs are the most common. They use the network card's MAC address (which unless spoofed, should be unique), plus a timestamp, plus the usual bit-twiddling to generate the UUID. In the case of a machine that doesn't have a MAC address, the 6 node bytes are generated with a cryptographically secure random number generator. If two UUIDs are generated in sequence fast enough that the timestamp matches the previous UUID, the timestamp is incremented by 1. Collisions should not occur unless one of the following happens: The MAC address is spoofed. One machine running two different UUID generating applications produces UUIDs at the exact same moment. Two machines without a network card or without user level access to the MAC address are given the same random node sequence, and generate UUIDs at the exact same moment. We run out of bytes to represent the timestamp and rollover back to zero.
Realistically, none of these events occur by accident within a single application's ID space. Unless you're accepting IDs on, say, an Internet-wide scale, or with an untrusted environment where malicious individuals might be able to do something bad in the case of an ID collision, it's just not something you should worry about. It's critical to understand that if you happen to generate the same version 4 UUID as I do, in most cases, it doesn't matter. I've generated the ID in a completely different ID space from yours. My application will never know about the collision so the collision doesn't matter. Frankly, in a single application space without malicious actors, the extinction of all life on earth will occur long before you have a collision, even on a version 4 UUID, even if you're generating quite a few UUIDs per second.
Also, 2^64 * 16 is 256 exabytes. As in, you would need to store 256 exabytes worth of IDs before you had a 50% chance of an ID collision in a single application space.