  This survey explores the development of adjoint Monte Carlo methods for
solving optimization problems governed by kinetic equations, a common challenge
in areas such as plasma control and device design. These optimization problems
are particularly demanding due to the high dimensionality of the phase space
and the randomness in evaluating the objective functional, a consequence of
using a forward Monte Carlo solver. To overcome these difficulties, a range of
``adjoint Monte Carlo methods'' have been devised. These methods skillfully
combine Monte Carlo gradient estimators with PDE-constrained optimization,
introducing innovative solutions tailored for kinetic applications. In this
review, we begin by examining three primary strategies for Monte Carlo gradient
estimation: the score function approach, the reparameterization trick, and the
coupling method. We also delve into the adjoint-state method, an essential
element in PDE-constrained optimization. Focusing on applications in the
radiative transfer equation and the nonlinear Boltzmann equation, we provide a
comprehensive guide on how to integrate Monte Carlo gradient techniques within
both the optimize-then-discretize and the discretize-then-optimize frameworks
from PDE-constrained optimization. This approach leads to the formulation of
effective adjoint Monte Carlo methods, enabling efficient gradient estimation
in complex, high-dimensional optimization problems.
