  Symmetry breaking beyond a global U(1) phase is the key signature of
unconventional superconductors. As prototypical strongly correlated materials,
heavy-fermion metals provide ideal platforms for realizing unconventional
superconductivity. In this article, we review heavy-fermion superconductivity,
with a focus on those materials with multiple superconducting phases. In this
context, we highlight the role of orbital-selective (matrix) pairing functions,
which are defined as matrices in the space of effective orbital degrees of
freedom such as electronic orbitals and sublattices as well as equivalent
descriptions in terms of intra- and inter-band pairing components in the band
basis. The role of quantum criticality and the associated strange-metal physics
in the development of unconventional superconductivity is emphasized
throughout. We discuss in some detail the recent experimental observations and
theoretical perspectives in the illustrative cases of UTe2, CeRh2As2, and
CeCu2Si2, where applied magnetic fields or pressure induce a variety of
superconducting phases. We close by providing a brief overview of overarching
issues and implications for possible future directions.
