Recent determinations of cosmological parameters point to a flat Universe, whose total energy density is composed of about two-thirds vacuum energy and one-third matter. Ordinary baryonic matter is relegated to a small fraction of the latter, within which the luminous part is an order of magnitude smaller yet. Particle dark matter, i.e., one or more relic particle species from the big bang, is thus strongly suggested as the dominant component of matter in the Universe. The axion, a hypothetical elementary pseudoscalar arising from the Peccei-Quinn solution to the strong-$\mathrm{CP}$ problem, is a well-motivated candidate. If the axion exists, it must be extremely light, in the mass range of ${10}^{\ensuremath{-}6}--{10}^{\ensuremath{-}3}\mathrm{eV},$ and possess extraordinarily feeble couplings to matter and radiation. Nevertheless, as proposed by Sikivie in 1983, the axion's two-photon coupling lends itself to a feasible search strategy with currently available technology. In this scheme, axions resonantly convert to single microwave photons by a Primakoff interaction, in a tunable microwave cavity permeated by a strong magnetic field. Present experiments utilizing heterostructure transistor microwave amplifiers have achieved total system noise temperatures of $\ensuremath{\sim}3\mathrm{K}$ and represent the world's quietest spectral radio receivers. Exclusion regions have already been published well into the band of realistic axion model couplings, within the lowest decade of mass range. Recent breakthroughs in the development of near-quantum-limited superconducting quantum interference device amplifiers should reduce the system noise temperature to $\ensuremath{\sim}100\mathrm{mK}$ or less. Ongoing research into using Rydberg-atom single-quantum detectors as the detector in a microwave cavity experiment could further reduce the effective noise temperature. In parallel with improvements in amplifier technology, promising concepts for higher-frequency cavity resonators are being explored to open up the higher decades in mass range. Definitive experiments to find or exclude the axion may therefore be at hand in the next few years. As the microwave cavity technique measures the total energy of the axion, a positive discovery could well reveal fine structure of the signal due to flows of nonthermalized axions. Manifesting diurnal and sidereal modulation, such detailed features would contain a wealth of information about the history, structure, and dynamics of our Milky Way galaxy.