Topics in Orbit Equivalence This volume provides a self contained introduction to some topics in orbit equivalence theory a branch of ergodic theory. The first two chapters focus on hyperfiniteness and amenability. Included here are proofs of Dyes theorem that probability measure preserving ergodic actions of the integers are orbit equivalent and of the theorem of Connes Feldman Weiss identifying amenability and hyperfiniteness for non singular equivalence relations. The presentation here is often influenced by descriptive set theory and Borel and generic analogs of various results are discussed. The final chapter is a detailed account of Gaboriaus recent results on the theory of costs for equivalence relations and groups and its applications to proving rigidity theorems for actions of free groups