Research in quantum computing has offered many new physical insights as well as the potential of exponentially increasing the computational power that can be harnessed to solve important problems in science and technology. 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 which, by their nature, limit errors. In this talk, I will discuss how to engineer topological superconductors 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 hosts Majorana zero modes. The defects binding these modes obey exotic (non-Abelian) exchange statistics. In this talk, I will review recent experimental efforts in realizing and detecting Majorana zero-energy modes in one-dimensional nanowires, and will discuss the progress towards building the first topological qubit.