In the past few years magneto-optical flux imaging (MOI) has come to take an increasing role in the investigation and understanding of critical current densities in high-T_c superconductors (HTS). This has been related to the significant progress in quantitative high-resolution magneto-optical imaging of flux distributions together with the model-independent determination of the corresponding current distributions. We review in this article the magneto-optical imaging technique and experiments on thin films, single crystals, polycrystalline bulk ceramics, tapes and melt-textured HTS materials and analyse systematically the properties determining the spatial distribution and the magnitude of the supercurrents. First of all, the current distribution is determined by the sample geometry. Due to the boundary conditions at the sample borders, the current distribution in samples of arbitrary shape splits up into domains of nearly uniform parallel current flow which are separated by current domain boundaries, where the current streamlines are sharply bent. Qualitatively, the current pattern is described by the Bean model; however, changes due to a spatially dependent electric field distribution which is induced by flux creep or flux flow have to be taken into account. For small magnetic fields, the Meissner phase coexists with pinned vortex phases and the geometry-dependent Meissner screening currents contribute to the observed current patterns. The influence of additional factors on the current domain patterns are systematically analysed: local magnetic field dependence of j_c(B), current anisotropy, inhomogeneities and local transport properties of grain boundaries. We then continue to an overview of the current distribution and current-limiting factors of materials, relevant to technical applications like melt-textured samples, coated conductors and tapes. Finally, a selection of magneto-optical experiments which give direct insight into vortex pinning and depinning mechanisms are reviewed.