The article provides an overview of the rapidly developing field of Majorana zero modes (MZMs) in solid-state systems, emphasizing their theoretical prediction, experimental realization, and potential use in topological quantum computation (TQC). MZMs are fermionic operators that commute with the Hamiltonian and exhibit non-Abelian braiding statistics, which are crucial for unitary gate operations in TQC. Recent experiments have shown specific signatures consistent with the existence of MZMs localized at the ends of semiconductor nanowires in the presence of superconducting proximity effect. The article discusses the experimental findings and their theoretical analyses, highlighting the extent to which these observations indicate the existence of anyonic MZMs in solid-state systems. It also explores fractional quantum Hall systems (the 5/2 state) and proposed schemes for carrying out braiding with MZMs, as well as the necessary steps for implementing TQC. The article reviews the current state of the field, including the challenges and potential future directions for research.The article provides an overview of the rapidly developing field of Majorana zero modes (MZMs) in solid-state systems, emphasizing their theoretical prediction, experimental realization, and potential use in topological quantum computation (TQC). MZMs are fermionic operators that commute with the Hamiltonian and exhibit non-Abelian braiding statistics, which are crucial for unitary gate operations in TQC. Recent experiments have shown specific signatures consistent with the existence of MZMs localized at the ends of semiconductor nanowires in the presence of superconducting proximity effect. The article discusses the experimental findings and their theoretical analyses, highlighting the extent to which these observations indicate the existence of anyonic MZMs in solid-state systems. It also explores fractional quantum Hall systems (the 5/2 state) and proposed schemes for carrying out braiding with MZMs, as well as the necessary steps for implementing TQC. The article reviews the current state of the field, including the challenges and potential future directions for research.