Since the introduction of the concept of topology into condensed matter physics, the investigation of topological materials has been a thriving subject in physics. Described by various kinds of topological invariants, these materials could host novel electronic and optical properties arising from their topological nature, such as conducting surface states in topological insulators and chiral magnetic effects in Weyl semimetals. Among these materials, zinc blende and wurtzite III-V materials are especially promising materials for people to study these topological properties, not only due to the large spin-orbit coupling strength of Bi, which can be used to induce band inversion and topological phase transitions, but also because they could be realized in experiments. However, most previous work in this area studies the band structures of GaAs, GaSb, and InSb; few work focuses on zinc blende and wurtzite structure GaBi and InBi. Therefore, this Capstone project uses first-principles calculation approaches to study the topological properties of zinc blende and wurtzite structure GaAs, GaSb, InSb, GaBi, and InBi, and achieves to identify zinc blende GaAs0.5Bi0.5as strong topological insulators, wurtzite GaBi and InBi as Dirac-Weyl semimetal, and wurtzite GaAs0.5Bi0.5 as triple point semimetals. This work could enrich the family of topological materials which can be realized in experiments, and their novel electronic properties arising from topology could also offer new possibilities to integrate them with semiconductors which have different functionalities together for the next-generation electronic, magnetic, optical, and quantum computing devices.