  We review recent progress on the instabilities of black strings and branes
both for pure Einstein gravity as well as supergravity theories which are
relevant for string theory. We focus mainly on Gregory-Laflamme instabilities.
In the first part of the review we provide a detailed discussion of the
classical gravitational instability of the neutral uniform black string in
higher dimensional gravity. The uniform black string is part of a larger phase
diagram of Kaluza-Klein black holes which will be discussed thoroughly. This
phase diagram exhibits many interesting features including new phases,
non-uniqueness and horizon-topology changing transitions. In the second part,
we turn to charged black branes in supergravity and show how the
Gregory-Laflamme instability of the neutral black string implies via a
boost/U-duality map similar instabilities for non- and near-extremal smeared
branes in string theory. We also comment on instabilities of D-brane bound
states. The connection between classical and thermodynamic stability, known as
the correlated stability conjecture, is also reviewed and illustrated with
examples. Finally, we examine the holographic implications of the
Gregory-Laflamme instability for a number of non-gravitational theories
including Yang-Mills theories and Little String Theory.
