  High precision spectroscopy can provide a sensitive tool to test Coulomb's
law on atomic length scales. This can then be used to constrain particles such
as extra "hidden" photons or minicharged particles that are predicted in many
extensions of the standard model, and which cause small deviations from
Coulomb's law. In this paper we use a variety of transitions in atomic
hydrogen, hydrogenic ions, and exotic atoms to probe Coulomb's law. This
extends the region of pure Coulomb's law tests to larger masses. For hidden
photons and minicharged particles this region is already tested by other
astrophysical and laboratory probes. However, future tests of true muonium and
muonic atoms are likely to probe new parameter space and therefore have good
discovery potential for new physics. Finally, we investigate whether the
discrepancy between the theoretical calculation of the 2s_{1/2}^{F=1} -
2p_{3/2}^{F=2} transition in muonic hydrogen and its recent experimental
measurement at PSI can be explained by the existence of a hidden photon. This
explanation is ruled out by measurements of the Lamb shift in ordinary
hydrogen.
