  We review matter wave and clock comparison tests of the gravitational
redshift. To elucidate their relationship to tests of the universality of free
fall (UFF), we define scenarios wherein redshift violations are coupled to
violations of UFF ("type II"), or independent of UFF violations ("type III"),
respectively. Clock comparisons and atom interferometers are sensitive to
similar effects in type II and precisely the same effects in type III
scenarios, although type III violations remain poorly constrained. Finally, we
describe the "Geodesic Explorer," a conceptual spaceborne atom interferometer
that will test the gravitational redshift with an accuracy 5 orders of
magnitude better than current terrestrial redshift experiments for type II
scenarios and 12 orders of magnitude better for type III.
