Recent results obtained through the 2D or 3D numerical simulation of the rise of magnetic flux tubes in the solar convection zone are discussed. The one--dimensional approximation known as the slender flux tube approximation permits calculation of the time evolution of a rising magnetic tube through most of the phases before it erupts at the surface. However, the 1D model disregards the off-axis structure of the magnetic tube, which turns out to play a non-negligible role in the tube evolution from the very beginning of its rise. This is discussed on the basis of a few examples: if the tube is untwisted, the hydrodynamical forces of the surrounding flow may easily convert it into a vortex tube pair whose components, asymptotically, stop to rise. If the tube is sufficiently twisted, then the development of vorticity is prevented in most of the tube interior, and the tube rises in a way reminiscent of air bubbles in a liquid. This suggests that the magnetic flux may be transported to the photosphere by means of twisted magnetic tubes. The physical conditions under which the magnetic flux is stored may be decisive for this issue: if the magnetic tubes are stored in mechanical equilibrium, then the minimum degree of twist required to prevent the generation of vorticity in the tube may be reduced. This review centers on the evolution of the magnetic flux in a tube-like geometry. However, recent 2D and 3D simulations of the time evolution of a Rayleigh-Taylor unstable magnetic slab are briefly discussed.