The bridge degree bdeg   v and cycle degree cdeg   v of a vertex v in a graph G are, respectively, the number of bridges and number of cycle edges incident with v in G . A characterization of finite nonempty sets S of nonnegative integers is given for which S is the set of bridge degrees (cycle degrees) of the vertices of some graph. The bridge-cycle degree of a vertex v in a graph G is the ordered pair ( b , c ) , where bdeg   v = b and cdeg   v = c . Those finite sets S of ordered pairs of nonnegative integers for which S is the set of bridge-cycle degrees of the vertices of some graph are also characterized.