We extend our previous investigations on the band structure of superlattices by applying the envelope-function approximation to four distinct problems. We calculate the band structure of HgTe-CdTe superlattices and show that these materials can be either semiconducting or zero-gap semiconductors, i.e., behave exactly like the ternary Hg1-xCdx Te random alloys. We analyze the superlattice dispersion relations in the layer planes (Landau superlattice subbands) and we compare the longitudinal and transverse effective masses of semiconducting InAs-GaSb superlattices. We calculate the general equation for the bound states due to aperiodic layers, taking account of the band structure of the host materials. We finally derive the dispersion relations of polytype (ABC or ABCD) superlattices.