This review article discusses recent advances in the search for Majorana fermions in solid-state systems, highlighting the potential of topological superconductors and heterostructures to engineer such particles. Majorana fermions, which are their own antiparticles, have significant implications for neutrino physics, dark matter searches, and quantum computing. The article begins by introducing exotic topological one- and two-dimensional superconductors that support Majorana fermions at their boundaries and vortices. It then explores the concept of "engineering" these exotic superconductors using heterostructures with ordinary s-wave superconductors, discussing various materials such as topological insulators, conventional semiconductors, and ferromagnetic metals. The experimental detection of Majorana fermions through tunneling, Josephson effects, and interferometry is addressed, emphasizing three classes of measurements that provide definitive evidence. Finally, the article discusses the non-Abelian exchange statistics of Majorana fermions and their potential for quantum computation, including the realization of fault-tolerant topological quantum computation schemes.This review article discusses recent advances in the search for Majorana fermions in solid-state systems, highlighting the potential of topological superconductors and heterostructures to engineer such particles. Majorana fermions, which are their own antiparticles, have significant implications for neutrino physics, dark matter searches, and quantum computing. The article begins by introducing exotic topological one- and two-dimensional superconductors that support Majorana fermions at their boundaries and vortices. It then explores the concept of "engineering" these exotic superconductors using heterostructures with ordinary s-wave superconductors, discussing various materials such as topological insulators, conventional semiconductors, and ferromagnetic metals. The experimental detection of Majorana fermions through tunneling, Josephson effects, and interferometry is addressed, emphasizing three classes of measurements that provide definitive evidence. Finally, the article discusses the non-Abelian exchange statistics of Majorana fermions and their potential for quantum computation, including the realization of fault-tolerant topological quantum computation schemes.