  Majorana fermions are spin-1/2 neutral particles that are their own
antiparticles and were initially predicted by Ettore Majorana in particle
physics but their observation still remains elusive. The concept of Majorana
fermions has been borrowed into condensed matter physics, where, unlike
particle physics, Majorana fermions emerge as zero-energy quasiparticles that
can be engineered by combining electrons and holes and have therefore been
coined Majorana zero modes. In this review, we provide a pedagogical
explanation of the basic properties of Majorana zero modes in unconventional
superconductors and their consequences in experimental observables, putting a
special emphasis on the initial theoretical discoveries. In particular, we
first show that Majorana zero modes are self-conjugated and emerge as a special
type of zero energy surface Andreev bound states at the boundary of
unconventional superconductors. We then explore Majorana zero modes in
one-dimensional spin-polarized $p$-wave superconductors, where we address the
formation of topological superconductivity and the physical realization in
superconductor-semiconductor hybrids. In this part we highlight that Majorana
quasiparticles appear as zero-energy edge states, exhibiting charge neutrality,
spin-polarized, and spatial nonlocality as unique properties that can be
already seen from their energies and wavefunctions. Next, we discuss
analytically obtained Green's functions of $p$-wave superconductors and
demonstrate that the emergence of Majorana zero modes is always accompanied by
the formation of odd-frequency spin-triplet pairing as a unique result of the
self-conjugate nature of Majorana zero modes. We finally address the signatures
of Majorana zero modes in tunneling spectroscopy, including the anomalous
proximity effect, and the phase-biased Josephson effect.
