The Heisenberg-Kitaev model is a paradigmatic model to describe the magnetism in honeycomb-lattice Mott insulators with strong spin-orbit coupling, such as A_{2}IrO_{3} (A=Na, Li) and α-RuCl_{3}. Here, we study in detail the physics of the Heisenberg-Kitaev model in an external magnetic field. Using a combination of Monte Carlo simulations and spin-wave theory, we map out the classical phase diagram for different directions of the magnetic field. Broken SU(2) spin symmetry renders the magnetization process rather complex, with sequences of phases and metamagnetic transitions. In particular, we find various large-unit-cell and multi-Q phases including a vortex-crystal phase for a field in the [111] direction. We also discuss quantum corrections in the high-field phase.