The Landau equation is one of the main equations in plasma physics, like the Boltzmann equation, it models the distribution of particle states by accounting both for transport effects and particle collisions. It’s analysis initially motivated the study of the Fokker-Planck equation, which represents a simplified model. The question of finite time blow up for the equation in the space-homogeneous regime has been a well-known open problem for many decades. In a recent preprint with Luis Silvestre we answer this question by showing the Fisher information is a Lyapunov functional for the equation, a fact that rules out blow ups. In this talk I will review the history of the Landau and related equations, discuss some of the breakthroughs of the last decade,  and discuss some of the ideas that made it possible to rule out blow ups, including the introduction of an auxiliary equation in double the number of variables, and a log-Sobolev inequality for functions on the sphere.