The 1937 theoretical discovery of Majorana fermions—particles that are their own antiparticles—has had a profound impact on various areas of physics, including neutrino physics, dark matter searches, the fractional quantum Hall effect, and superconductivity. Despite this long history, the unambiguous observation of Majorana fermions remains a significant challenge. This review highlights recent advances in the search for Majorana fermions in solid-state systems, which have led many researchers to believe that this goal may soon be achieved. The article begins by introducing exotic topological one- and two-dimensional superconductors that support Majorana fermions at their boundaries and vortices. It then discusses how such superconductors can be engineered in the laboratory using heterostructures with ordinary s-wave superconductors. Various materials, including topological insulators, conventional semiconductors, and ferromagnetic metals, are considered for this purpose. The article addresses the experimental detection of Majorana fermions, focusing on three key methods: tunneling, Josephson effects, and interferometry. These methods provide clear signatures of Majorana fermions. The article also discusses the remarkable properties of condensed matter Majorana fermions, particularly their non-Abelian exchange statistics, which have potential applications in quantum computing. Majorana fermions in solid-state systems are not fundamental particles but emergent excitations. They arise in superconductors where Cooper pairs condense, leading to zero-energy modes at the ends of one-dimensional topological p-wave superconductors and at vortices in two-dimensional p+ip superconductors. These zero-modes are the condensed matter realization of Majorana fermions. The article explains that Majorana fermions exhibit non-Abelian statistics, a property that is crucial for quantum computing. This is because the exchange of Majorana fermions leads to non-commutative transformations of the wavefunction, enabling fault-tolerant quantum computation. The article also discusses the potential for experimental realization of these phenomena, emphasizing the importance of topological phases and the role of non-Abelian statistics in quantum information processing. The review concludes by highlighting the significance of these developments for both fundamental physics and technological applications, particularly in the quest for a scalable quantum computer. The article underscores the importance of topological phases and the potential for experimental realization of Majorana fermions in solid-state systems.The 1937 theoretical discovery of Majorana fermions—particles that are their own antiparticles—has had a profound impact on various areas of physics, including neutrino physics, dark matter searches, the fractional quantum Hall effect, and superconductivity. Despite this long history, the unambiguous observation of Majorana fermions remains a significant challenge. This review highlights recent advances in the search for Majorana fermions in solid-state systems, which have led many researchers to believe that this goal may soon be achieved. The article begins by introducing exotic topological one- and two-dimensional superconductors that support Majorana fermions at their boundaries and vortices. It then discusses how such superconductors can be engineered in the laboratory using heterostructures with ordinary s-wave superconductors. Various materials, including topological insulators, conventional semiconductors, and ferromagnetic metals, are considered for this purpose. The article addresses the experimental detection of Majorana fermions, focusing on three key methods: tunneling, Josephson effects, and interferometry. These methods provide clear signatures of Majorana fermions. The article also discusses the remarkable properties of condensed matter Majorana fermions, particularly their non-Abelian exchange statistics, which have potential applications in quantum computing. Majorana fermions in solid-state systems are not fundamental particles but emergent excitations. They arise in superconductors where Cooper pairs condense, leading to zero-energy modes at the ends of one-dimensional topological p-wave superconductors and at vortices in two-dimensional p+ip superconductors. These zero-modes are the condensed matter realization of Majorana fermions. The article explains that Majorana fermions exhibit non-Abelian statistics, a property that is crucial for quantum computing. This is because the exchange of Majorana fermions leads to non-commutative transformations of the wavefunction, enabling fault-tolerant quantum computation. The article also discusses the potential for experimental realization of these phenomena, emphasizing the importance of topological phases and the role of non-Abelian statistics in quantum information processing. The review concludes by highlighting the significance of these developments for both fundamental physics and technological applications, particularly in the quest for a scalable quantum computer. The article underscores the importance of topological phases and the potential for experimental realization of Majorana fermions in solid-state systems.