We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered. In one of the most general formulations, a graph G = (V, E), sets L(v) of admissible colors, and natural numbers c v for the vertices v ∈ V are given, and the question is

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... whether there can be chosen a subset C(v) ⊆ L(v) of cardinality c v for each vertex in such a way that the sets C(v), C(v ) are disjoint for each pair v, v of adjacent vertices. The particular case of constant |L(v)| with c v = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite these two invariants have nearly the same value for almost all graphs. To illustrate typical techniques, some of the proofs are sketched.