  We collect some of our favorite proofs of Brooks' Theorem, highlighting
advantages and extensions of each. The proofs illustrate some of the major
techniques in graph coloring, such as greedy coloring, Kempe chains, hitting
sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex
coloring, such as list coloring, online list coloring, and Alon--Tarsi
orientations, since analogues of Brooks' Theorem hold in each context. We
conclude with two conjectures along the lines of Brooks' Theorem that are much
stronger, the Borodin--Kostochka Conjecture and Reed's Conjecture.
