Whereas a rational map of the Riemann sphere has finitely many critical points, the critical points of an endomorphism of P^n generally constitute an irreducible hypersurface. Moreover, the dynamics of this critical locus under iteration differs starkly from the one-dimensional case. In this talk I will discuss what's known and unknown about the critical locus, including my own contributions, joint work with Max Weinreich, and recent results of Luo--Meng and Broustet--Höring. There will be plenty of examples!
