The Fossil Record Indicates...
The oldest fossils that people don't disagree about are at about 3 billion or so years ago. Strictly, you see things like stromatolites with layers of bacteria. Stromatolites are, oh, they're defined as organo-sedimentary laminated structures. Basically, the idea is that they're fossils that are communities and the communities show this sort of fine lamination spliced in them. And they do involve organisms and also inorganic deposits. What I'll show you in a minute is that we have modern anologs of these ancient stromatolites.
So this is what you see in the fossil record way back there. The origin of eukaryotes, I put it at about 2 billion years ago. It may be much earlier than that, it may be a little bit later. I suspect it's earlier but I can't tell you for sure. The problem is that when you look at the fossil records to show that it's a eukaryote, you start to look at issues like size and so on, and there is no question that eukaryotes' size range overlap the prokaryotes. So we're actually applying a very stringent definition. So really the eukaryotes could be much earlier.