13 Let S = (a1, a2, . . .) be an infinite nondecreasing sequence of positive 14 integers. An S-packing k-coloring of a graph G is a mapping from V (G) to 15 {1, 2, . . . , k} such that vertices with color i have pairwise distance greater 16 than ai, and the S-packing chromatic number χS(G) of G is the smallest 17 integer k such that G has an S-packing k-coloring. This concept general18 izes the concept of proper coloring (when S = (1, 1, 1, . . .)) and broadcast 19 coloring (when S = (1, 2, 3, 4, . . .)). In this paper, we consider bounds on 20 the parameter and its relationship with other parameters. We characterize 21 the graphs with χS = 2 and determine χS for several common families of 22 graphs. We examine χS for the infinite path and give some exact values and 23 asymptotic bounds. Finally we consider complexity questions, especially 24 about recognizing graphs with χS = 3. 25