The dynamical Mordell-Lang conjecture says that if V is a variety over a field of characteristic 0, f: V -> V is a morphism, W is a subvariety of V, and z is a point on V, then the set of n such that f^n(z) is in W should form a finite union of arithmetic sequences. This may be seen a generalization of the Skolem-Mahler-Lech theorem for linear recurrence sequences and of the cyclic case of the classical Mordell-Lang conjecture (now a theorem of Vojta, Faltings, and McQuillan). The dynamical Mordell-Lang conjecture remains open. We will present a survey of known results and techniques around this problem, and then discuss several related questions, including questions about integral points in orbits, points in value sets in orbits, and a conjectured dynamical analog of the Serre open image theorem.