We report electronic and structural properties of cubic and hexagonal 3C-, 2H-, 4H-, and 6H-SiC bulk crystals and of the C-terminated SiC(001)-c(2×2) surface as resulting from density functional theory (DFT) within local density approximation (LDA). In particular, we employ newly constructed nonlocal, norm-conserving pseudopotentials which incorporate self-interaction corrections. Results obtained with usual pseudopotentials show the typical LDA shortcomings, most noticeably the systematic underestimate of the band gap. These problems are attributed to an unphysical self-interaction inherent in the common DFT-LDA. We describe the construction of appropriate self-interaction-corrected pseudopotentials for Si and C atoms and show how they can be transferred to the SiC solid by adequate modifications. It is in the very nature of our pseudopotentials that they cause no additional computational effort, as compared to usual pseudopotentials in standard LDA calculations. To test their transferability to different crystal structures we apply these pseudopotentials to both cubic and hexagonal polytypes of SiC. The resulting energy gaps are in excellent agreement with experimental data and the bulk band structures are in most gratifying agreement with the results of considerably more elaborate quasiparticle calculations. Structural properties of the different polytypes are found in excellent agreement with experiment, as well, not showing the usual LDA underestimate of lattice constants and overestimate of bulk moduli. Also the electronic structure of SiC(001)-c(2×2), calculated to exemplify the usefulness of the pseudopotentials for surfaces, shows improved agreement with experiment as compared to the respective surface band structure obtained within standard LDA.