Let f:X-->X be a rational map in complex dimension two. Assuming the first dynamical degree of f exceeds 1, there is a standard way to construct a dynamically natural positive closed current invariant under pullback. The construction requires, however, that f be `algebraically stable.' In this talk I'll focus on some recent examples where the dynamical degree is transcendental and construct invariant currents in a situation where algebraic stability is unachievable.