Symplectic geometry and combinatorics are strongly intertwined due to the existence of Hamiltonian torus actions. These actions are associated with a special map (called the moment map) which "transforms" a compact symplectic manifold into a convex polytope. We will concentrate on the  special class of reflexive polytopes which was introduced by Batyrev in the context of mirror symmetry and  has attracted much attention recently. In particular, we will see how the famous "12 and 24" properties in dimension 2 and 3 can be generalized with the help of symplectic geometry.