  We review recent studies of contact and thermodynamic geometry for black
holes in AdS spacetimes in the extended thermodynamics framework. The
cosmological constant gives rise to the notion of pressure $P = -\Lambda / 8
\pi$ and, subsequently a conjugate volume $V$, thereby leading to a close
analogy with hydrostatic thermodynamic systems. To begin with, we review the
contact geometry approach to thermodynamics in general and then consider
thermodynamic metrics constructed as the Hessians of various thermodynamic
potentials. We then study their correspondence to statistical ensembles for
systems with two-dimensional spaces of equilibrium states. From the zeroes and
divergences of the curvature scalar obtained from the metric, we carefully
analyze the issue of ensemble non-equivalence and show certain complimentary
behaviors in the description of a thermodynamic system. Following a thorough
analysis of the familiar van der Waals system, we turn our attention to black
holes in extended phase space. Considering the example of charged AdS black
holes, we discuss the generic features of their thermodynamic geometry in
detail. The relationship of the thermodynamic curvature(s) with critical points
as well as microscopic interactions in black holes is also briefly explored. We
finally set up the thermodynamic geometry for finite temperature gauge theories
dual to black holes in AdS via holographic correspondence and comment on recent
progress.
