The Rabin-Keisler theorem and the sizes of ultrapowers

Recall the Rabin-Keisler theorem which gives a lower bound κω for the size of proper elementary extensions of complete structures of size κ, provided that κ is an infinite cardinal below the first measurable cardinal. We survey – and at places clarify and extend – some facts which connect the Rabin-Keisler theorem, sizes of ultrapowers, combinatorial properties of ultrafilters, and large cardinals.