Majorana fermions, particles that are their own antiparticles, can exist in condensed matter systems as bound states of electron and hole excitations in superconductors. They can be constructed using a superconductor to hide charge differences and a topological (Berry) phase to eliminate energy differences. A pair of Majorana fermions bound to magnetic or electrostatic defects exhibits non-Abelian exchange statistics, making them promising for quantum computing due to their long coherence times. This review discusses methods to detect Majorana fermions in topological superconductors and their potential applications in quantum computers. It covers the theoretical foundations, experimental strategies, and recent developments in the search for Majorana fermions. The review highlights the importance of topological phases, the role of spin-orbit coupling, and the use of superconducting and magnetic materials to realize Majorana fermions. It also discusses detection methods such as half-integer conductance quantization, nonlocal tunneling, the 4π-periodic Josephson effect, and thermal metal-insulator transitions. The review concludes with an outlook on the experimental progress in realizing Majorana fermions, emphasizing the potential of semiconductor nanowires and topological insulators in this endeavor.Majorana fermions, particles that are their own antiparticles, can exist in condensed matter systems as bound states of electron and hole excitations in superconductors. They can be constructed using a superconductor to hide charge differences and a topological (Berry) phase to eliminate energy differences. A pair of Majorana fermions bound to magnetic or electrostatic defects exhibits non-Abelian exchange statistics, making them promising for quantum computing due to their long coherence times. This review discusses methods to detect Majorana fermions in topological superconductors and their potential applications in quantum computers. It covers the theoretical foundations, experimental strategies, and recent developments in the search for Majorana fermions. The review highlights the importance of topological phases, the role of spin-orbit coupling, and the use of superconducting and magnetic materials to realize Majorana fermions. It also discusses detection methods such as half-integer conductance quantization, nonlocal tunneling, the 4π-periodic Josephson effect, and thermal metal-insulator transitions. The review concludes with an outlook on the experimental progress in realizing Majorana fermions, emphasizing the potential of semiconductor nanowires and topological insulators in this endeavor.