  The subject of diffraction of waves by sharp boundaries has been studied
intensively for well over a century, initiated by groundbreaking mathematicians
and physicists including Sommerfeld, Macdonald and Poincar\'e. The significance
of such canonical diffraction models, and their analytical solutions, was
recognised much more broadly thanks to Keller, who introduced a geometrical
theory of diffraction (GTD) in the middle of the last century, and other
important mathematicians such as Fock and Babich. This has led to a very wide
variety of approaches to be developed in order to tackle such two and three
dimensional diffraction problems, with the purpose of obtaining elegant and
compact analytic solutions capable of easy numerical evaluation.
  The purpose of this review article is to showcase the disparate mathematical
techniques that have been proposed. For ease of exposition, mathematical
brevity, and for the broadest interest to the reader, all approaches are aimed
at one canonical model, namely diffraction of a monochromatic scalar plane wave
by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The
first three approaches offered are those most commonly used today in
diffraction theory, although not necessarily in the context of wedge
diffraction. These are the Sommerfeld-Malyuzhinets method, the Wiener-Hopf
technique, and the Kontorovich-Lebedev transform approach. Then follows three
less well-known and somewhat novel methods, which would be of interest even to
specialists in the field, namely the embedding method, a random walk approach,
and the technique of functionally-invariant solutions.
  Having offered the exact solution of this problem in a variety of forms, a
numerical comparison between the exact solution and several powerful
approximations such as GTD is performed and critically assessed.
