In generalizing the classical theory of circle maps, we study the rotation set for maps of the real line x↦f(x) with almost periodic displacement f(x) - x. Such maps are in one-to-one correspondence with maps of compact Abelian topological groups with the displacement taking values in a dense one-parameter subgroup. For homeomorphisms, we show the existence of the analogue of the Poincaré rotation number, which is the common rotation number of all orbits besides possibly those that have rotation zero. (The coexistence of zero and non-zero rotation numbers is the main new phenomenon compared with the classical circle case.) For non-invertible maps, we prove results concerning the realization of points of the rotation interval as the rotation numbers of orbits and ergodic measures. We also address the issue of practical computation of the rotation number.