  I review the concept of a {\em disorder operator}, introduced originally by
Kadanoff in the context of the two-dimensional Ising model. Disorder operators
acquire an expectation value in the disordered phase of the classical spin
system. This concept has had applications and implications to many areas of
physics ranging from quantum spin chains to gauge theories to topological
phases of matter. In this paper I describe the role that disorder operators
play in our understanding of ordered, disordered and topological phases of
matter. The role of disorder operators, and their generalizations, and their
connection with dualities in different systems, as well as with Majorana
fermions and parafermions, is discussed in detail. Their role in recent
fermion-boson and boson-boson dualities is briefly discussed.
