We discuss recent results on frustration of quantum charges in lattice models for itinerant fermions with strong repulsive interactions. P. Fendley and K. Schoutens introduced a class of models that, as a result of a judicious tuning of kinetic and potential terms, possess supersymmetry. In 1D this model is solved analytically and turns out to be quantum critical. The thermodynamic limit is described by an N=2 superconformal field theory. In 2D the model exhibits superfrustration: an extensive degeneracy of supersymmetric ground states. Using techniques from cohomology the ground state degeneracy can be obtained analytically. We demonstrate how for the 2D square lattice the ground state counting problem is fully solved through a remarkable correspondence with specific rhombus tilings of the plane.