In this talk I present classification results for structures B that are first-order definable over a countably infinite homogeneous structure. Those results resemble the situation in the finite: either 3SAT has a primitive positive interpretation in an expansion of (the model-complete core of) B by finitely many constants, or B has a four-ary polymorphism that satisfies the Siggers term identities ”up to automorphisms”. In the first case, CSP(B) is NP-hard by general principles. In the second case, one would expect that CSP(B) can be solved in polynomial-time, and indeed this has been verified for all structures B that are definable over the order of the rationals, the Rado graph, the universal homogeneous C-relation, or the equivalence relation with infinitely many infinite classes.