The review provides a pedagogical explanation of Majorana zero modes (MZMs) in unconventional superconductors, emphasizing their theoretical discoveries. MZMs are spin-1/2 particles that are their own antiparticles, initially predicted by Ettore Majorana in particle physics but observed in condensed matter physics as zero-energy quasiparticles. The review covers the basic properties of MZMs, their emergence as zero-energy surface Andreev bound states (ZESABSs) at the boundary of unconventional superconductors, and their formation in one-dimensional (1D) spin-polarized $p$-wave superconductors. It highlights that MZMs exhibit unique properties such as charge neutrality, spin polarization, and spatial nonlocality. The review also discusses the analytical Green's functions of $p$-wave superconductors, the formation of odd-frequency spin-triplet pairing, and experimental signatures of MZMs, including tunneling spectroscopy, anomalous proximity effect, and phase-biased Josephson effect. The authors aim to provide a comprehensive understanding of MZMs and their potential applications in quantum technologies.The review provides a pedagogical explanation of Majorana zero modes (MZMs) in unconventional superconductors, emphasizing their theoretical discoveries. MZMs are spin-1/2 particles that are their own antiparticles, initially predicted by Ettore Majorana in particle physics but observed in condensed matter physics as zero-energy quasiparticles. The review covers the basic properties of MZMs, their emergence as zero-energy surface Andreev bound states (ZESABSs) at the boundary of unconventional superconductors, and their formation in one-dimensional (1D) spin-polarized $p$-wave superconductors. It highlights that MZMs exhibit unique properties such as charge neutrality, spin polarization, and spatial nonlocality. The review also discusses the analytical Green's functions of $p$-wave superconductors, the formation of odd-frequency spin-triplet pairing, and experimental signatures of MZMs, including tunneling spectroscopy, anomalous proximity effect, and phase-biased Josephson effect. The authors aim to provide a comprehensive understanding of MZMs and their potential applications in quantum technologies.