AbstractMotivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every ‐degenerate graph is also weakly ‐degenerate. On the other hand, if is weakly ‐degenerate, then (and, moreover, the same bound holds for the list‐chromatic and even the DP‐chromatic number of ). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly 4‐degenerate, which implies Thomassen's famous theorem that planar graphs are 5‐list‐colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph of maximum degree is weakly ‐degenerate unless . (By contrast, all ‐regular graphs have degeneracy .) We actually prove an even stronger result, namely that for every , there is such that if is a graph of weak degeneracy at least , then either contains a ‐clique or the maximum average degree of is at least . Finally, we show that graphs of maximum degree and either of girth at least 5 or of bounded chromatic number are weakly ‐degenerate, which is best possible up to the value of the implied constant.