Key subjects related to the present status of mantle convection theory are reviewed in this paper. Rheology of the polycrystalline mantle material is known from laboratory experiments. Diffusion and dislocation creeps must predominate in the long-term deformation of the mantle at high temperatures; their effective viscosities can be estimated from measured creep parameters. Inhomogeneities in the chemical and phase composition of the mantle can influence the convective pattern considerably. Perhaps the most significant heterogeneities in this respect are those produced by the phase changes of the transition zone. The endothermic spinel-perovskite transition at 660 km depth can create an efficient obstacle to vertical flow. Available seismic tomographic images show clear signs of such an obstacle in some subduction zones, at other places however the downwellings seem to be unimpeded. The exact degree of flow layering caused by the 660 km discontinuity is not known, but some sort of flow stratification is strongly suggested by the isotopic and trace element geochemistry which shows that different chemical reservoirs must exist in the mantle on the geological time scale. Equations and the main governing forces are analyzed, and the basic structures of the convective flow are demonstrated in examples of numerical solutions. In particular, recent modelling results are discussed with regard to the plume forms allowed by a semi-permeable internal phase boundary. It is shown that three different kinds of plumes can reach the surface and produce hotspots: those arising from the internal phase boundary, those coming from the basal boundary layer of the mantle, and a recently described new plume type which breaks through the phase transition starting from a diffuse volume below the transition zone.