Majorana fermions are spin-1/2 neutral particles that are their ownantiparticles and were initially predicted by Ettore Majorana in particlephysics but their observation still remains elusive. The concept of Majoranafermions has been borrowed into condensed matter physics, where, unlikeparticle physics, Majorana fermions emerge as zero-energy quasiparticles thatcan be engineered by combining electrons and holes and have therefore beencoined Majorana zero modes. In this review, we provide a pedagogicalexplanation of the basic properties of Majorana zero modes in unconventionalsuperconductors and their consequences in experimental observables, putting aspecial emphasis on the initial theoretical discoveries. In particular, wefirst show that Majorana zero modes are self-conjugated and emerge as a specialtype of zero energy surface Andreev bound states at the boundary ofunconventional superconductors. We then explore Majorana zero modes inone-dimensional spin-polarized $p$-wave superconductors, where we address theformation of topological superconductivity and the physical realization insuperconductor-semiconductor hybrids. In this part we highlight that Majoranaquasiparticles appear as zero-energy edge states, exhibiting charge neutrality,spin-polarized, and spatial nonlocality as unique properties that can bealready seen from their energies and wavefunctions. Next, we discussanalytically obtained Green's functions of $p$-wave superconductors anddemonstrate that the emergence of Majorana zero modes is always accompanied bythe formation of odd-frequency spin-triplet pairing as a unique result of theself-conjugate nature of Majorana zero modes. We finally address the signaturesof Majorana zero modes in tunneling spectroscopy, including the anomalousproximity effect, and the phase-biased Josephson effect.