In this thesis, various canonical problems of plane wave diffraction by infinite two-dimensional wedges are studied in both acoustic and electromagnetic physical settings. The thesis is divided into two main parts. The first, which is essentially an extensive review of extant methods, focuses on wedge diffraction with homogeneous Dirichlet or Neumann boundary conditions. It involves studying the well-known solution to Sommerfeld's half-plane diffraction problem and provides an extensive review of the literature on the perfect wedge problem, including analytic methods such as the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques, as well as asymptotic techniques such as Keller's geometrical theory of diffraction. The second part of the thesis is dedicated to the problem of diffraction by a penetrable wedge. To this day, there is no clear analytical solution to this important canonical problem, but there have been numerous attempts at computational and asymptotic solutions by many reputable authors using extensions of techniques applied to perfect wedge diffraction. Because it is penetrable, the material properties between the wedge scatterer and the exterior host differ. It is hence possible to define the so-called contrast parameter as the ratio of specific material properties (depending on the physical context) between the host and the scatterer. Throughout the thesis, this parameter is considered to be small and the contrast is said to be high. This assumption allows construction of an asymptotic iterative scheme, which enables the penetrable wedge problem to be written as an infinite sequence of impenetrable wedge problems. All but the first of these impenetrable wedge problems are solved using a combination of the Sommerfeld-Malyuzhinets and Wiener-Hopf techniques. The result is a sequence of complex nested integrals which are evaluated using a subtle interplay of interpolation, asymptotic expansions and advanced complex analysis. For several test cases, including those illustrated in this thesis (that were chosen for mathematical convenience and consistency with the literature), the numerical results were in good agreement with alternative approaches.