Are Penrose tilings universal? Do aperiodic universal tilings exist?

I thought that EVERY plane tiling by the two Penrose tiles is universal (later I checked, that is indeed the case as Peter explains nicely below). It is cheap but you could take any (set of) possible animal(s) and call them new tiles. I think that Penrose started with a set of 6 tiles, probably with the same property (and probably derivable in the way I mentioned from the two tile set). You could also split one or both Penrose tiles into weird pieces which only fit together to recreate that piece.

If I recall correctly, it is an open question if there is a single tile which only tiles aperiodically. SO it is premature to ask if there is one which has a universal tiling (not that you asked that).

In the periodic case (from when that was part of the question) I can certainly specify two (or even any k) gawky tiles which only tile in one way so that is automatically universal. It is easier to think of colored tiles. (maybe squares constrained to fit 4 at each corner) it is easy to see how to make a universal tiling. There are ways to replace these with huge squarish polyomonoes with lock and key bumps.

A very nice reference on this family of questions is Grunbaum and Shephard's book Tilings and Patterns (now out in paperback). It has starred problems which are hard, and double-starred problems which are research projects. It is in this book that we find the problem of whether there is a tile which only tiles aperiodically. (There is such a beast for the 3-dimensional analog, but it's a bit of a cheat.) I contacted Grunbaum about this a few years ago, and he said the problem was still open.