The surface subgroup question asks for the existence of a subgroup of a given group H which is isomorphic to the fundamental group of a closed surface of genus at least 2. We begin with a survey on recent progress on this question. We then explain the strategy to construct such surface subgroups in any cocompact lattice of a classical simple Lie group of non-compact type different from SO(n,1) for even n. The construction is explicit and geometric and extends earlier work of Kahn and Markovic and of Kahn, Labourie, Mozes.