  We review the theory and applications of complex stochastic quantization to
the quantum many-body problem. Along the way, we present a brief overview of a
number of ideas that either ameliorate or in some cases altogether solve the
sign problem, including the classic reweighting method, alternative
Hubbard-Stratonovich transformations, dual variables (for bosons and fermions),
Majorana fermions, density-of-states methods, imaginary asymmetry approaches,
and Lefschetz thimbles. We discuss some aspects of the mathematical
underpinnings of conventional stochastic quantization, provide a few
pedagogical examples, and summarize open challenges and practical solutions for
the complex case. Finally, we review the recent applications of complex
Langevin to quantum field theory in relativistic and nonrelativistic quantum
matter, with an emphasis on the nonrelativistic case.
