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.
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