The properties, applications, and growth of the wide bandgap semiconductors GaN and SiC will be described. With their relatively large bandgaps of about 3 eV, high bond strengths and high thermal conductivities, these materials have found a range of applications for bulk-to-UV optical devices and for high speed / high power electronics. We have studied the growth of GaN on SiC substrates using plasma-assisted molecular beam epitaxy. Careful preparation and nucleation procedures permits the growth of films having a relatively low density of threading dislocations of about 1x10⁹cm^-2, nearly all of which have edge character. Using a combination of scanning tunneling microscopy and first principles theory the surfaces of the films are found to be terminated by 2 monolayers of metallic Ga. This terminating layer is beneficial for enhancing surface diffusivity, but it is detrimental in that the Ga fills dislocation cores. Growth in the presence of H is found to solve this problem, with the dislocation cores being terminated by H rather than metallic Ga. In an effort to reduce dislocation density, growth has been performed on porous SiC substrates with the aim of achieving lateral growth over the pores. Results were not promising, although use of thin porous SiN interlayers is found to achieve significant dislocation reduction. Finally, work will be described in which the porous SiC substrates are used as semipermeable membranes for the sensing of proteins in a biotechnology application.