Epitaxial growth of SiC on hexagonal (or α)-SiC(0001) has been performed by means of solid-source molecular beam epitaxy (MBE). The solid-source MBE growth conditions have been analyzed concerning the supersaturation and the excess phase formation of silicon and carbon. In general, our results demonstrate that control of the Si/C ratio and the supersaturation (S) is essential for the growth mode and the kind of polytype grown. Low temperature (T<1450K) deposition on on-axis SiC substrates always results in the growth of 3C-SiC, which is significantly improved by an alternating supply of Si and C. On vicinal substrates, a step flow growth mode has been realized at T down to 1300K. In experiments performed at T>1450K under near surface equilibrium conditions, different growth modes, and conditions stabilizing the growth of certain polytypes have been found. With a step decrease of S, a step-flow growth mode of both 4H-and 6H-SiC was obtained and, for the first time in case of epitaxial SiC growth, a homogeneous nucleation of α-SiC at more C-rich conditions has been realized. Conditions stabilizing the growth of certain polytypes have been estimated by thermodynamic calculations considering the influence of polytype structure on the supersaturation and the surface energy. Based on these results, we have demonstrated the growth of a double-heterostructure by firstly growing a 3C-SiC layer on 4H-SiC(0001) at low temperature and a subsequent growth of 4H-SiC under near surface equilibrium conditions on a C-stabilized surface on top of this layer.