John Platt has a nice summary of recent activity in the are of de-extinction. This is where you use modern genetic techniques to bring species that are extinct back into existence.
I find it interesting that casual talk about this sort of thing almost always starts out with things like de-extinction very large and very long extinct, and I’m sure, very expensive to take care of creatures like dinosaurs or wooly mammoths. People in the de-extinction business (and there are some, and there have even been some efforts carried out) are more realistic, of course.
I’ve always said we should start by cloning something that is not extinct, from its remains. Start with the dumpster behind a KFC and see if we can get a chicken. (If it turns out to not be a chicken, that’s another matter.) After doing that a few times, try cloning something that has a living ecological analog: The Quagga, for instance. They went extinct recently and are basically a variant of a zebra (though a different species). Then if that works look into endemic recently extinct animals such as the dodo.
After that, we can sit down and talk mammoths and passenger pigeons.