A red-white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and the [Formula: see text]th coordinate of the code is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The problem of determining those graphs that are ID-graphs is investigated. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text] and is denoted by ID([Formula: see text]). It is shown that (1) a nontrivial connected graph [Formula: see text] has ID-number 1 if and only if [Formula: see text] is a path, (2) the path of order 3 is the only connected graph of diameter 2 that is an ID-graph, and (3) every positive integer [Formula: see text] different from 2 can be realized as the ID-number of some connected graph. The identification spectrum of an ID-graph [Formula: see text] is the set of all positive integers [Formula: see text] such that [Formula: see text] has an ID-coloring with exactly [Formula: see text] red vertices. Identification spectra are determined for paths and cycles.