Compositionally graded group IIa-fluoride layers are applied as epitaxial buffers in semiconductor heteroepitaxy to overcome large lattice and thermal expansion mismatch. We review our work on composites with lattice mismatch up to -20%, like CdTe-BaFJCaF2 on Si and PbSnSe-BaF2/CaF2-Si, including some applications and of nearly lattice-and thermal-expansion-matched BaF2 on PbSe. New results on the growth mode of the fluoride layers on Si using in situ Rheed and Auger analysis will be included and the unique thermal mismatch strain relief capabilities of the fluoride buffers discussed.