The existence of an event horizon is the defining property of a black hole. I shall review the properties of event horizons and discuss various examples demonstrating that event horizons are not smooth in dynamical processes such as black hole formation or merger. I shall explain how non-smooth features of an event horizon can be classified into various types, such as creases and caustics. I shall classify ``perestroikas'' of these structures, in which they undergo a qualitative change at an instant of time. A crease perestroika gives an exact local description of the event horizon near the ``instant of merger'' of a black hole merger. Other crease perestroikas describe event horizon nucleation or collapse of a hole in a toroidal horizon. Caustic perestroikas provide a mechanism for smoothing the horizon. I shall suggest that creases make a contribution to black hole entropy.