  In this article, we review the current state of the art in using levitated
nanoparticles to answer questions related to thermodynamics and non-equilibrium
physics. We begin in Section 2 with a summary of the relevant deterministic and
stochastic forces, which determine the particle dynamics and allow for control
of the particle. In Section 3 we give a brief review of the stochastic (i.e.
Brownian) motion of levitated particles, since the Brownian particle is
fundamental to the theory of stochastic thermodynamics. Then we discuss the
stability of the trapped particles and the related Kramers escape problem in
Section 4. After that, we introduce effective potentials for the energy in
Section 5. These potentials are useful to describe the dynamics of levitated
nanoparticles in a time-modulated trap, where the particle is driven far away
from equilibrium. In Section 6 we discuss the dynamics of relaxation towards
equilibrium, before we review the work on fluctuation theorems in Section 7.
Fluctuation theorems are a powerful generalization of the well known
thermodynamic inequalities for systems far from equilibrium. Finally, Section 8
discusses the potential of constructing new kinds of heat engines based on
nanoparticles levitated in high vacuum.
