  We present an in-depth exploration of the phenomenon of dynamical friction in
a universe where the dark matter is composed entirely of so-called Fuzzy Dark
Matter (FDM), ultralight bosons of mass $m\sim\mathcal{O}(10^{-22})\,$eV. We
review the classical treatment of dynamical friction before presenting analytic
results in the case of FDM for point masses, extended mass distributions, and
FDM backgrounds with finite velocity dispersion. We then test these results
against a large suite of fully non-linear simulations that allow us to assess
the regime of applicability of the analytic results. We apply these results to
a variety of astrophysical problems of interest, including infalling satellites
in a galactic dark matter background, and determine that \emph{(1)}~for FDM
masses $m\gtrsim 10^{-21}\, {\rm eV}\, c^{-2}$, the timing problem of the
Fornax dwarf spheroidal's globular clusters is no longer solved and
\emph{(2)}~the effects of FDM on the process of dynamical friction for
satellites of total mass $M$ and relative velocity $v_{\rm rel}$ should require
detailed numerical simulations for $\left(M/10^9~M_{\odot}\right)
\left(m/10^{-22}~{\rm eV}\right)\left(100~{\rm km}~{\rm s}^{-1}/v_{\rm
rel}\right) \sim 1$, parameters which would lie outside the validated range of
applicability of any currently developed analytic theory, due to transient wave
structures in the time-dependent regime.
