Research in quantum computing has offered many important new physical insights as well as the potential of exponentially increasing the computational power that can be harnessed to solve important problems in energy, medicine, computer science, physics, mathematics and material science. The largest fundamental barrier to building a scalable quantum computer is errors caused by decoherence. Topological quantum computing evades this barrier by exploiting topological materials that by their nature limit errors. One of the main challenges in condensed matter physics is the exploration of topological phases of matter that could be used to build a scalable quantum computer. Among the exciting recent developments in this direction are the discoveries of new phases of matter with many intriguing properties such as topological insulators and superconductors. In this talk, I will focus on topological superconductors and discuss how one can engineer non-trivial superconductivity in the laboratory at the interface of a conventional superconductor and a semiconductor with spin-orbit interaction. I will show that such a topological state emerging at the interface can be occupied by exotic quasi-particles, called Majorana fermions, that obey unconventional exchange statistics. Their unique properties can be exploited for implementing fault-tolerant topological quantum computation schemes that are inherently decoherence-free. I will conclude my talk by proposing several experiments for detecting Majorana fermions in one-dimensional nanowires.