> I do seem to remember being taught in school that the definition of species is "organisms that can reproduce and generate fertile offspring".

Which is problematic for extinct animals. We have no way of knowing if modern humans would be able to produce offspring with Neanderthals, for example.

That's a problematic criterion when trying to classify long extinct species. We can hypothesize, and perhaps there is a point where we can confidently say "These two definitely couldn't mate" (e.g. Australopithecus afarensis and Homo neanderthalensis), but there's always going to be a large gray area. In fact, here's an interesting quote from the OP article:

> A small group of Homo sapiens left Africa 60,000 years ago and settled the rest of the world, replacing the other human species they encountered (with a small amount of interbreeding).

So actually, it seems we are confident that Homo sapiens was capable of reproduction with some of the other "species" we've identified.

It's easy to see why it's a problem when we look into the past. Presumably, every creature should be considered to the be the same species as its parents. But go far enough back and your N-great grandparents are undeniably a different species.

If we had a record of every organism that had ever lived, trying to classify them into "species" would be a ridiculous task -- regardless of how punctuated-equilibriumy things are. Since the fossil record is fairly spotty, it's only a slightly ridiculous task.

Yeah, not that simple at all. To me though, the problem isn't the difficulty of defining the species. The problem is peoples' need to classify things (a problem that goes far beyond scientific classifications). But I recognize that I'm in the minority in that view.