  This review discusses the electronic properties and the prospective research
directions of superconductor-graphene heterostructures. The basic electronic
properties of graphene are introduced to highlight the unique possibility of
combining two seemingly unrelated physics, superconductivity and relativity. We
then focus on graphene-based Josephson junctions, one of the most versatile
superconducting quantum devices. The various theoretical methods that have been
developed to describe graphene Josephson junctions are examined, together with
their advantages and limitations, followed by a discussion on the advances in
device fabrication and the relevant length scales. The phase-sensitive
properties and phase-particle dynamics of graphene Josephson junctions are
examined to provide an understanding of the underlying mechanisms of Josephson
coupling via graphene. Thereafter, microscopic transport of correlated
quasiparticles produced by Andreev reflections at superconducting interfaces
and their phase-coherent behaviors are discussed. Quantum phase transitions
studied with graphene as an electrostatically tunable two-dimensional platform
are reviewed. The interplay between proximity-induced superconductivity and the
quantum-Hall phase is discussed as a possible route to study topological
superconductivity and non-Abelian physics. Finally, a brief summary on the
prospective future research directions is given.
