  Study of the hadronic matrix elements can provide not only tests of the QCD
sector of the Standard Model (in comparing with existing experiments) but also
reliable low-energy hadronic quantities applicable to a wide range of
beyond-the-Standard Model scenarios where experiments or theoretical
calculations are limited or difficult. On the QCD side, progress has been made
in the notoriously difficult problem of addressing gluonic structure inside the
nucleon, reaching higher-$Q^2$ region of the form factors, and providing a
complete picture of the proton spin. However, even further study and
improvement of systematic uncertainties are needed. There are also proposed
calculations of higher-order operators in the neutron electric dipole moment
Lagrangian, which would be useful when combined with effective theory to probe
BSM. Lattice isovector tensor and scalar charges can be combined with upcoming
neutron beta-decay measurements of the Fierz interference term and neutrino
asymmetry parameter to probe new interactions in the effective theory,
revealing the scale of potential new TeV particles. Finally, I revisit the
systematic uncertainties in recent calculations of $g_A$ and review prospects
for future calculations.
