Recent advances in coupling superconductivity to near surface InAs quantum wells (QWs) have allowed the observation of zero bias peaks suggestive of Majorana Zero Modes on a scalable platform[1]. This paves the way for lithographically defined complex nanostructured networks necessary for topological quantum computation. A semiconductor with high spin-orbit coupling should lead to a large induced topological gap and hence improved topological protection. Our efforts have focused on developing high mobility of near surface quantum wells of the high spin-orbit semiconductors InAs [2] and InSb. We have also investigated the growth of InAsySb1-y, which is expected to have higher spin-orbit coupling than either InAs or InSb. Epitaxial Al films are grown in-situ at low temperature so as to minimize interfacial reactions and Al agglomeration on the surface [3]. We have found that due to the high surface mobility of Al even at room temperature, the Al surface must be passivated immediately after growth to prevent agglomeration. Rather than relying on post growth lithography and top down etching to form semiconductor nanowire structures, we have investigated epitaxial Al growth on atomic hydrogen cleaned MOVPE-grown vapor-liquid-solid InSb nanostructures[4] and in-vacuum chemical beam epitaxy selective area grown InAs nanowires[5]. Large zero bias peaks have been observed for these epitaxial Al/InSb nanowire structures[6]. In this presentation progress, as well as issues, in developing superconductor/III-V semiconductor heterostructures for topological quantum computer application will be discussed.