This volume consists of the written presentations of lectures given at two special sessions: the AMS Special Session on Topology in Dynamics (Winston-Salem, NC) and the AMS-AWM Special Session on Geometry in Dynamics (San Antonio, TX). Each article concerns aspects of the topology or geometry of dynamical systems. Topics covered include the following: foliations and laminations, iterated function systems, the three-body problem, isotopy stability, homoclinic tangles, fractal dimension, Morse homology, knotted orbits, inverse limits, contact structures, Grassmanians, blowups, and continua.