We present a theoretical investigation of how n- and p-type doping affect the band structure around the band gap of 3C-, 2H-, 4H-, and 6H-SiC. For comparison we also consider Si. We have calculated for various values of the dopant concentration (i) the shift in energy of the bottom (top) of the conduction (valence) band, (ii) the band gap narrowing, (iii) the shift of the optical band gap, and (iv) the doping-induced changes in conduction band curvature, i.e., changes in effective electron masses in n-type materials. In addition we have also (v) estimated the critical concentration for Mott transitions and (vi) calculated the shifts in conduction- and valence bands caused, not by doping, but by injection of an electron-hole plasma of various concentrations. To study the effects of doping we have considered a system consisting of impurity ions immersed in a (high-density) gas of majority carriers and a low-density gas of minority carriers. The changes in the bands relative to the idealised crystal are then regarded as being due to interparticle Coulomb interactions and associated particle correlation in and between the gases, as well as to electron and hole interactions with the randomly distributed ions. We have considered two models. The simplest model for band edge displacements is analytical and based on relatively simple assumptions like parabolic energy bands and simple modelling of electron correlation effects. The second model is numerical and includes full band non-parabolicity, and the electron and hole gas interactions are treated in the random-phase approximation.