  Over the last decade, numerical solutions of Quantum Chromodynamics (QCD)
using the technique of lattice QCD have developed to a point where they are
beginning to connect fundamental aspects of nuclear physics to the underlying
degrees of freedom of the Standard Model. In this review, the progress of
lattice QCD studies of nuclear matrix elements of electroweak currents and
beyond-Standard-Model operators is summarized, and connections with effective
field theories and nuclear models are outlined.
  Lattice QCD calculations of nuclear matrix elements can provide guidance for
low-energy nuclear reactions in astrophysics, dark matter direct detection
experiments, and experimental searches for violations of the symmetries of the
Standard Model, including searches for additional CP violation in the hadronic
and leptonic sectors, baryon-number violation, and lepton-number or flavor
violation. Similarly, important inputs to neutrino experiments seeking to
determine the neutrino-mass hierarchy and oscillation parameters, as well as
other electroweak and beyond-Standard-Model processes can be determined. The
phenomenological implications of existing studies of electroweak and
beyond-Standard-Model matrix elements in light nuclear systems are discussed,
and future prospects for the field toward precision studies of these matrix
elements are outlined.
