It is fairly obvious that any hashing algorithm that is used to map any number of arbitrary length pieces of data to some fixed size number, will produce collisions--lots of them. In fact, an infinite number of collisions!

Not trying to split hairs, but I would say a finite, large number but not infinite. There is a difference: if it was infinite you could keep finding collisions forever but if it is finite there would be a time that you would stop finding collisions.

And I don't think that it is overstated. It has often been said that there are collisions in MD5. Its strength was that it took a long time and/or alot of computing power to find the thing that generates a particular hash which i will call plaintext. Now that time is lessened. It doesn't matter if one finds the original plain text or an alternative the generates the same hash the computer will not know the difference. eg MD5 is used for passwords on linux machines. No matter what if the plaintext at logon matches the hash in the shadow file login will be granted whether it was "the original plaintext" or a "collision plaintext"

Update:See posts by BrowserUk and adrianh below. I was thinking the collisions were restricted to the set of possible hashes. Bizarre!