The article reviews the progress in realizing topological superconductivity and Majorana zero modes (MZMs) in semiconductor-superconductor (SC-SC) heterostructures. MZMs, which are neutral excitations in a superconductor, have significant implications for quantum information processing due to their non-Abelian exchange statistics and topological protection. The review discusses the material science advancements, such as the growth of high-mobility semiconductors and the preparation of high-quality interfaces between superconductors and semiconductors. It also covers robust signatures of MZMs, including zero-bias tunneling conductance measurements and Coulomb blockade experiments, which provide strong evidence for the presence of MZMs in proximitized nanowires. The article concludes by discussing the prospects for implementing MZMs-based topological quantum computation.The article reviews the progress in realizing topological superconductivity and Majorana zero modes (MZMs) in semiconductor-superconductor (SC-SC) heterostructures. MZMs, which are neutral excitations in a superconductor, have significant implications for quantum information processing due to their non-Abelian exchange statistics and topological protection. The review discusses the material science advancements, such as the growth of high-mobility semiconductors and the preparation of high-quality interfaces between superconductors and semiconductors. It also covers robust signatures of MZMs, including zero-bias tunneling conductance measurements and Coulomb blockade experiments, which provide strong evidence for the presence of MZMs in proximitized nanowires. The article concludes by discussing the prospects for implementing MZMs-based topological quantum computation.