  We review recent progress in taking the large dimension limit of Einstein's
equations. Most of our analysis is classical in nature and concerns situations
where there is a black hole horizon although we briefly discuss various
extensions that include quantum gravitational effects. The review consists of
two main parts: the first a discussion of general aspects of black holes and
effective membrane theories in this large dimension limit, and the second a
series of applications of this limit to interesting physical problems. The
first part includes a discussion of quasinormal modes which leads naturally
into a description of effective hydrodynamic-like equations that describe the
near horizon geometry. There are two main approaches to these effective
theories -- a fully covariant approach and a partially gauge-fixed one -- which
we discuss in relation to each other. In the second part we divide the
applications up into three main categories: the Gregory-Laflamme instability,
black hole collisions and mergers, and the anti-de Sitter/conformal field
theory correspondence (AdS/CFT). AdS/CFT posits an equivalence between a
gravitational theory and a strongly interacting field theory, allowing us to
extend our spectrum of applications to problems in hydrodynamics, condensed
matter physics, and nuclear physics. A final, shorter part of the review
describes further promising directions where there have been, as yet, few
published research articles.
