  Over the past fifteen years, tremendous efforts have been devoted to
realizing topological superconductivity in realistic materials and systems,
predominately propelled by their promising application potentials in
fault-tolerant quantum information processing. In this article, we attempt to
give an overview on some of the main developments in this field, focusing in
particular on two-dimensional crystalline superconductors that possess either
intrinsic p-wave pairing or nontrivial band topology. We first classify the
three different conceptual schemes to achieve topological superconductor (TSC),
enabled by real-space superconducting proximity effect, reciprocal-space
superconducting proximity effect, and intrinsic TSC. Whereas the first scheme
has so far been most extensively explored, the subtle difference between the
other two remains to be fully substantiated. We then move on to candidate
intrinsic or p-wave superconductors, including Sr2RuO4, UTe2, Pb3Bi, and
graphene-based systems. For TSC systems that rely on proximity effects, the
emphases are mainly on the coexistence of superconductivity and nontrivial band
topology, as exemplified by transition metal dichalcogenides, cobalt pnictides,
and stanene, all in monolayer or few-layer regime. The review completes with
discussions on the three dominant tuning schemes of strain, gating, and
ferroelectricity in acquiring one or both essential ingredients of the TSC, and
optimizations of such tuning capabilities may prove to be decisive in our drive
towards braiding of Majorana zero modes and demonstration of topological
qubits.
