The field of time-domain astrophysics has entered the era of Multi-messenger Astronomy (MMA). One key science goal for the next decade (and beyond) will be to characterize gravitational wave (GW) and neutrino sources using the next generation of Extremely Large Telescopes (ELTs). These studies will have a broad impact across astrophysics, informing our knowledge of the production and enrichment history of the heaviest chemical elements, constrain the dense matter equation of state, provide independent constraints on cosmology, increase our understanding of particle acceleration in shocks and jets, and study the lives of black holes in the universe. Future GW detectors will greatly improve their sensitivity during the coming decade, as will near-infrared telescopes capable of independently finding kilonovae from neutron star mergers. However, the electromagnetic counterparts to high-frequency (LIGO/Virgo band) GW sources will be distant and faint and thus demand ELT capabilities for characterization. ELTs will be important and necessary contributors to an advanced and complete multi-messenger network.