The study of geodesic flows on homogeneous spaces is an area of research that has recently yielded some fascinating developments. This book focuses on many of these, with one of its highlights an elementary and complete proof by Margulis and Dani of Oppenheim's conjecture. Other features are self-contained treatments of an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; and Ledrappier's example of a mixing action which is not a mixing of all orders.