Is the clique problem NP-complete also on bipartite or planar graphs?
In a bipartite graph there can be no clique of size three. Not much of a problem that is left I guess?

(edit) Sorry, that was too fast, I realize thanks to comment by Nicholas below. To make up for it I googled a reference.

The maximum edge biclique problem is NP-complete by Ren*e Peeters. Discrete Applied Mathematics 131 (2003) 651 – 654

(ps) So I understand that my answer relates to a question that changes the types of graphs we are looking for, kind of bipartite cliques, whereas the other answer restricts the set of graphs we are looking in to planar. In planar graphs cliques can have size at most four, thanks to Kuratowski.