Fano varieties are projective varieties with “positive curvature”. Examples of Fano varieties are projective spaces, products of projective spaces, Grassmannians and hypersurfaces in projective spaces of low degree. Recently, K-stability (which is the algebro-geometric version of Kähler-Einstein metrics in complex-differential geometry) has been used to construct compact (actually projective) moduli spaces of Fano varieties. In this talk I will explain how to use combinatorial methods (e.g. Minkowski sums of polytopes and toric geometry) to produce singular points on moduli spaces of Fano varieties. This is based on work with Anne-Sophie Kaloghiros.