  This review concerns recent results on the quantitative study of convergence
towards the stationary state for spatially inhomogeneous kinetic equations. We
focus on analytical results obtained by means of certain probabilistic
techniques from the ergodic theory of Markov processes. These techniques are
sometimes referred to as Harris-type theorems. They provide constructive proofs
for convergence results in the $L^1$ (or total variation) setting for a large
class of initial data. The convergence rates can be made explicit (both for
geometric and sub-geometric rates) by tracking the constants appearing in the
hypotheses. Harris-type theorems are particularly well-adapted for equations
exhibiting non-explicit and non-equilibrium steady states since they do not
require prior information on the existence of stationary states. This allows
for significant improvements of some already-existing results by relaxing
assumptions and providing explicit convergence rates. We aim to present
Harris-type theorems, providing a guideline on how to apply these techniques to
the kinetic equations at hand. We discuss recent quantitative results obtained
for kinetic equations in gas theory and mathematical biology, giving some
perspectives on potential extensions to nonlinear equations.
