  The Laser Interferometer Space Antenna (LISA) will be a transformative
experiment for gravitational wave astronomy, and, as such, it will offer unique
opportunities to address many key astrophysical questions in a completely novel
way. The synergy with ground-based and space-born instruments in the
electromagnetic domain, by enabling multi-messenger observations, will add
further to the discovery potential of LISA. The next decade is crucial to
prepare the astrophysical community for LISA's first observations. This review
outlines the extensive landscape of astrophysical theory, numerical
simulations, and astronomical observations that are instrumental for modeling
and interpreting the upcoming LISA datastream. To this aim, the current
knowledge in three main source classes for LISA is reviewed; ultracompact
stellar-mass binaries, massive black hole binaries, and extreme or intermediate
mass ratio inspirals. The relevant astrophysical processes and the established
modeling techniques are summarized. Likewise, open issues and gaps in our
understanding of these sources are highlighted, along with an indication of how
LISA could help making progress in the different areas. New research avenues
that LISA itself, or its joint exploitation with upcoming studies in the
electromagnetic domain, will enable, are also illustrated. Improvements in
modeling and analysis approaches, such as the combination of numerical
simulations and modern data science techniques, are discussed. This review is
intended to be a starting point for using LISA as a new discovery tool for
understanding our Universe.
