Explore the fascinating interplay between symmetry and asymmetry in dynamical systems through this 55-minute Joint AMS-MAA Invited Address delivered by Amie Wilkinson from the University of Chicago. Delve into the fundamental concepts of symmetry, dynamical symmetry, and the dynamical symmetry group. Discover how symmetries map orbits to orbits and preserve asymptotic dynamical invariants, illustrated with examples such as Julia sets. Examine smooth flows and their symmetries, emphasizing the special nature of symmetries in dynamics. Investigate Hamiltonian dynamics, Noether's Theorem, Smale's conjecture, and Kopell's Theorem. Conclude with an exploration of symmetry rigidity, featuring insights from a collaborative project with D. Damjanović and D. Xu.