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Given an integer $k>4$ and a graph $H$, we prove that, assuming P$\neq$NP,the List-$k$-Coloring Problem restricted to $H$-free graphs can be solved inpolynomial time if and only if either every component of $H$ is a path on atmost three vertices, or removing the isolated vertices of $H$ leaves an inducedsubgraph of the five-vertex path. In fact, the "if" implication holds for all$k\geq 1$.

List-$k$-Coloring $H$-free graphs for all $k>4$