Adaptive mesh-refinement (AMR) schemes for integration over two-dimensional Brillouin zones are pre-sented and their properties are investigated in detail. A salient feature of these integration techniques is that the grid of sampling points is automatically adapted to the integrand in such a way that regions with high accuracy demand are sampled with high density, while the other regions are sampled with low density. This adaptation may save a sizable amount of computation time in comparison with those integration methods without mesh refinement. Several AMR schemes for one- and two-dimensional integration are introduced. As an application, the spin-dependent conductance of electronic tunneling through planar junctions is investigated and discussed with regard to Brillouin zone integration.