The clique complex of a graph is a simplicial complex with a simplex for each clique. Clique complexes are frequently being computed in applications of topology to data, but we do not understand their algorithmic theory or their mathematical theory. I will introduce clique complexes of graphs, explain why applied topologists care about them, and survey open problems about the topology of clique complexes of unit disk graphs, powers of lattice graphs, and powers of hypercube graphs.