A projective algebraic variety is called Fano (after the Italian mathematician Gino Fano) if it has positive curvature. Fano varieties play a prominent rôle in algebraic geometry for many reasons. Recently there has been fundamental work on constructing moduli spaces of (certain) Fano varieties. The aim of my talk is to show how polytopes and combinatorics can help in proving that moduli spaces of Fano varieties are, in general, quite singular. My talk is based on joint work with Anne-Sophie Kaloghiros. The first 45 minutes of my talk do not require any knowledge in algebraic geometry.