Let G be a countable infinite group. Unless G is virtually abelian, the unitary dual of G (that is, the equivalence classes of irreducible unitary representations of G) is not a countably separated with respect to a natural Borel structure and its classification is therefore considered as being hopeless. A sensible substitute for the unitary dual is the set Prim(G) of equivalence classes of irreducible unitary representations for a weaker equivalence relation. The set Prim(G) is related to the space of characters Char(G) of G, which is a more concrete object and contains in general both finite as well as infinite characters. We will review all these duals spaces and describe some recent results concerning the case where G is the group of K-rational points of an algebraic group over an infinite field K.