Light axion-like particles occur in many theories of beyond-Standard-Model physics, and may make up some or all of the universe's dark matter. One of the ways they can couple to the Standard Model is through the electromagnetic $F_{����} \tilde F^{����}$ portal, and there is a broad experimental program, covering many decades in mass range, aiming to search for axion dark matter via this coupling. In this paper, we derive limits on the absorbed power, and coupling sensitivity, for a broad class of such searches. We find that standard techniques, such as resonant cavities and dielectric haloscopes, can achieve O(1)-optimal axion-mass-averaged signal powers, for given volume and magnetic field. For low-mass (frequency $\ll$ GHz) axions, experiments using static background magnetic fields generally have suppressed sensitivity - we discuss the physics of this limitation, and propose experimental methods to avoid it, such as microwave up-conversion experiments. We also comment on the detection of other forms of dark matter, including dark photons, as well as the detection of relativistic hidden sector particles.

27 pages, 3 figures; v5: matches version to appear in PRD