Let X be a variety defined over an algebraically closed field of characteristic 0 and let phi : X -> X be a birational automorphism. The Medvedev-Scanlon conjecture predicts when there is a rational point of X with dense orbit under phi. We prove their conjecture in positive Kodaira dimension and then, contingent on conjectures in the Minimal Model Program, prove the conjecture for certain minimal threefolds of Kodaira dimension 0. This is joint work with Jason Bell, Dragos Ghioca, and Zinovy Reichstein. Speaker(s): Matt Satriano (University of Waterloo)