Experimental and theoretical studies are presented of HgTe CdTe superlattices. We report the first study of the photoluminescence and photoconductivity spectra of superlattices. We find that Photoluminescence, photoconductivity and infrared absorption all point to the same band gap for the superlattice. Theoretical studies of these superlattices have given the first estimates of the magnitude of the optical absorption, the first study of the role of strain, and the first study of the role of the band offset in determining the properties of superlattices. We have also invented a novel electronic device that has negative resistance based on HgCdTe-CdTe-HgCdTe heterostructures. We have carried out the first theoretical studies of superlattices that include dilute magnetic semiconductors. Superlattices of CdTe-ZnTe have been studied experimentally and theoretically to give the first investigations of the properties of possible II-IV visible light emitters.