Majorana fermions, proposed more than seven decades ago by E. Majorana in the context of high energy physics (to describe neutrinos), are finally beginning to be experimentally realized in condensed matter systems. Following our recent proposal to realize them in semiconductor-superconductor heterostructures, at least four experimental groups worldwide have claimed to observe them in various experiments. In this talk I will introduce the physics of the Majorana fermions, why they are technologically important (topological quantum computation), our proposal for Majorana fermions in solid state systems, and the subsequent experimental realizations that have been reported.