Symmetry is one of those concepts that easily bridges the underlying themes of this conference: art, architecture and mathematics. Part of the fascination of Penrose tilings is their rather quirky kind of symmetry. But if one is pressed to describe exactly what this symmetry is, say group theoretically, one is likely to come up empty handed. Penrose tilings are aperiodic - they have no translational symmetries. They are not built upon simple repetition of a motif. Furthermore, in general Penrose tilings (there are infinitely many different ones) do not have 5-fold symmetry, though they have a strong feeling of "five-ness" about them. So in general they have no symmetries in the conventional way.