The study of convex polyhedra from a mathematical point of view began in ancient Greece well over two thousand years ago. It is believed that Euclid’s “Elements” was written to inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic solids”, as well as to place their study on a secure mathematical foundation. The discovery of convex polytopes (the analogous figures in four or more dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on this subject have been published. However, there are still many unsolved problems, and the purpose of this article is to state a few of these. These problems have widespread appeal because they can be readily understood with very little previous mathematical knowledge, and although they may be quite difficult, there is always the possibility that someone with a very minimum of mathematical background will devise a method of approach that leads to a solution.