Abstract: The Landau equation models the particle density of a diffuse plasma. Mathematically, the interaction between transport and nonlinear, nonlocal diffusion makes this a challenging model to analyze, and the question of global existence vs. breakdown remains unanswered in the general case. However, some recent breakthroughs have clarified this question in certain particular cases. In this talk, I will start by introducing the model and its mathematical difficulties, discuss some important recent progress, and finally outline some directions of ongoing work.