Summary: We present the general quantum-mechanical description of Dirac particles with electric and magnetic dipole moments in arbitrary electromagnetic, inertial and gravitational fields. The covariant Dirac equation, extended by the terms describing anomalous magnetic and electric dipole moments, is transformed to the Foldy-Wouthuysen representation. This allows to obtain the Schrödinger form of quantum-mechanical equations of motion and to establish the classical limit of relativistic quantum mechanics. We demonstrate the complete agreement between the quantum mechanics and the classical electrodynamics in Riemannian spacetimes in the general case. As an example, we consider the spin dynamics in a gravitational wave in the presence of electromagnetic fields.