A review of previous work on the transverse instability of solitary waves is given in this chapter with an emphasis on topics that resonate with themes in this monograph. The basic state throughout is a solitary wave that is localised and decays exponentially at infinity. Transverse instability of other types of solitary waves as well as periodic travelling waves, although related and important, will be only briefly mentioned. It is the spectral problem associated with the linearisation that is of main interest. Nonlinear stability is not considered, but some results that build on the linear stability problem are reviewed. The principal applications of the theory familiar to the authors are in optics and in the theory of water waves. The former are reviewed briefly in the nonlinear Schr’́odinger equation (NLS) section below, and the latter in the section on water waves. Attention will be restricted to PDEs that have been identified as Hamiltonian, or can be shown to be Hamiltonian (classical or multisymplectic).