Galactic binaries with orbital periods less than $approx$1 hr are strong gravitational wave sources in the mHz regime, ideal for the Laser Interferometer Space Antenna (LISA). In fact, theory predicts that emph{LISA} will resolve tens of thousands of Galactic binaries individually with a large fraction being bright enough for electromagnetic observations. This opens up a new window where we can study a statistical sample of compact Galactic binaries in both, the electromagnetic as well the gravitational wavebands. Using multi-messenger observations we can measure tidal effects in detached double WD systems, which strongly impact the outcome of WD mergers. For accreting WDs as well as NS binaries, multi-messenger observations give us the possibility to study the angular momentum transport due to mass transfer. In this white paper we present an overview of the opportunities for research on Galactic binaries using multi-messenger observations and summarize some recommendations for the 2020 time-frame.
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