We propose to search for axion dark matter via the oscillating electric dipole moments that axions induce in atoms and molecules. These moments are produced through the intrinsic oscillating electric dipole moments of nucleons and through the P, T-violating nucleon-nucleon interaction mediated by pion exchange, both of which arise due to the axion-gluon coupling, and also directly through the axion-electron interaction. Axion dark matter may also be sought for through the spin-precession effects that axions produce by directly coupling to fermion spins