A systematic analysis of low-frequency waves such as hydromagnetic and acoustic waves in a magnetized dusty plasma containing electrons, ions and dust particles is presented. Starting with the three-fluid equations and the Maxwell equations, we derive, retaining finite ion mass, an effective two-fluid model incorporating deviations from the frozen-in-fluid approximation for the ion and electron fluid motions. We show that normal modes exist in two widely separated frequency regimes. In addition to obtaining hydromagnetic waves that are generalizations of the usual Alfven and magneto-acoustic modes in a two-component electron–ion plasma, we demonstrate the existence of a new class of magneto-acoustic waves (both fast and slow type) called ‘dustmagneto-acoustic waves’. These modes have qualitatively different dependences on the equilibrium parameters such as density, magnetic field and temperature when compared with the usual magneto-acoustic waves. The new modes arise owing to finite ion mass (compared with the dust-particle mass), leading to an effective inertial resistivity that inhibits the ion (as well as the electron) fluid from being frozen to the magnetic field lines. The fast branch of the dust-magneto-acoustic waves is shown to be the electromagnetic generalization of the electrostatic dust-acoustic wave recently obtained in unmagnetized dusty plasmas. In the two-component limit the new modes degenerate into the usual type of magneto-acoustic waves. Dispersion relations for various other modes are also presented.