A polytope is the convex hull of finitely many points in a Euclidean space. A polytope is centrally symmetric if it is symmetric about the origin. In this talk we will concentrate on simplicial polytopes and their face numbers. While the face numbers of general simplicial polytopes are very well understood, the situation is drastically different once the symmetry assumption is added. This talk will be a survey of what is known and mostly of what is not know about face numbers of these fascinating polytopes. If time permits, we will also touch on the situation for simplicial spheres with and without symmetry.