  The AdS/CFT correspondence relates certain strongly coupled CFTs with large
effective central charge $c_\text{eff}$ to semi-classical gravitational
theories with AdS asymptotics. We describe recent progress in understanding
gravity duals for CFTs on non-trivial spacetimes at finite temperature, both in
and out of equilibrium. Such gravity methods provide powerful new tools to
access the physics of these strongly coupled theories, which often differs
qualitatively from that found at weak coupling.
  Our discussion begins with basic aspects of AdS/CFT and progresses through
thermal CFTs on the Einstein Static Universe and on periodically identified
Minkowski spacetime. In the latter context we focus on states describing
so-called plasma balls, which become stable at large $c_\text{eff}$. We then
proceed to out-of-equilibrium situations associated with dynamical bulk black
holes. In particular, the non-compact nature of these bulk black holes allows
stationary solutions with non-Killing horizons that describe time-independent
flows of CFT plasma. As final a topic we consider CFTs on black hole
spacetimes. This discussion provides insight into how the CFT transports heat
between general heat sources and sinks of finite size. In certain phases the
coupling to small sources can be strongly suppressed, resulting in negligible
heat transport despite the presence of a deconfined plasma with sizeable
thermal conductivity. We also present a new result, explaining how this
so-called droplet behaviour is related to confinement via a change of conformal
frame.
