  Depending on the exact experimental conditions, the thermodynamic properties
of physical systems can be related to one or more thermostatistical ensembles.
Here, we survey the notion of thermodynamic temperature in different
statistical ensembles, focusing in particular on subtleties that arise when
ensembles become non-equivalent. The 'mother' of all ensembles, the
microcanonical ensemble, uses entropy and internal energy (the most
fundamental, dynamically conserved quantity) to derive temperature as a
secondary thermodynamic variable. Over the past century, some confusion has
been caused by the fact that several competing microcanonical entropy
definitions are used in the literature, most commonly the volume and surface
entropies introduced by Gibbs. It can be proved, however, that only the volume
entropy satisfies exactly the traditional form of the laws of thermodynamics
for a broad class of physical systems, including all standard classical
Hamiltonian systems, regardless of their size. This mathematically rigorous
fact implies that negative 'absolute' temperatures and Carnot efficiencies $>1$
are not achievable within a standard thermodynamical framework. As an important
offspring of microcanonical thermostatistics, we shall briefly consider the
canonical ensemble and comment on the validity of the Boltzmann weight factor.
We conclude by addressing open mathematical problems that arise for systems
with discrete energy spectrum.
