Geometrodynamics: the nonlinear dynamics of curved spacetime

We review discoveries in the nonlinear dynamics of curved spacetime, largely made possible by numerical solutions of Einstein's equations. We discuss critical phenomena and self-similarity in gravitational collapse, the behavior of spacetime curvature near singularities, the instability of black strings in five spacetime dimensions, and the collision of four-dimensional black holes. We also discuss the prospects for further discoveries in geometrodynamics via observations of gravitational waves.