In 1937 Italian physicist Ettore Majorana predicted the existence of strange fermionic particles that are their own antiparticles. It is possible that neutrinos realize such Majorana fermions but 75 years after the historical prediction the evidence remains inconclusive. In this talk I will describe recent efforts to engineer and observe Majorana fermions in solid state systems which appear to be very close to fruition. Majorana fermions have been theoretically predicted to occur in a class of systems called topological superconductors. Although such systems do not seem to exist in nature they can be engineered by combining other ingredients such as the ordinary superconductors and semiconductors with strong spin-orbit coupling or topological insulators. Signatures consistent with Majorana fermions have already been reported in such hybrid devices and the race is on for the first conclusive experimental observation. I will explain the intriguing physics behind these solid-state realizations of Majorana fermions and discuss their significance for future technologies.