Geometric intersection graphs are intensively studied both for their practical motivations and interesting theoretical properties. Many classes allow elegant characterizations, for many of them optimization problems NP-hard in general can be solved in polynomial time. We will present a survey of recent results and old problems in this area, including questions related to colorability, maximum clique or representations of planar and co-planar graphs. Computational complexity of recognition of many of intersection defined classes of graphs will be one of the main topics.