  The linear scalar quantum field, propagating in a globally hyperbolic
spacetime, is a relatively simple physical model that allows us to study many
aspects in explicit detail. In this review we focus on the theory of thermal
equilibrium (KMS) states of such a field in a stationary spacetime. Our
presentation draws on several existing sources and aims to give a unified
exposition. We also take the opportunity to weaken some of the technical
assumptions of the earlier literature. In particular, we completely drop any
assumptions on the behaviour of the norm and lapse of the Killing field. This
is especially important for physical applications to the exterior region of a
stationary black hole.
  Our review includes results on the existence of a unique ground state and of
unique quasi-free KMS states, as well as an evaluation of the evidence that
motivates the use of the KMS-condition to characterise thermal equilibrium. We
especially draw attention to the poorly understood behaviour of the notion of
temperature of a quantum field with respect to locality.
  If the spacetime is standard static (i.e. if it admits a Cauchy surface that
is orthogonal to the Killing field) the analysis can be done even more
explicitly. For compact Cauchy surfaces we consider Gibbs states and their
properties. For general Cauchy surfaces we give a detailed and rigorous
justification of the technique of Wick rotation, including the explicit
determination of the Killing-time dependence of the quasi-free KMS states.
