An Eigenvalue solution of the Helmholtz Equation with Arbitrary Two-Dimensional Boundaries using Finite element Methods with Application to Electromagnetic Waveguides Beginning with Maxwell's equations, the Helmholtz equation in two dimensional geometry is derived and then applied to an infinite length uniform waveguide with arbitrary cross section. The transverse Electric (TE) and Transverse Magnetic (TM) waveguide modes are defined based on the eigenfunctions of the Helmholtz equation with both Dirichlet and Neumann boundary conditions. The superposition principle is employed to show that the TE and TM modes are complete, i.e., any arbitrary field in the prescribed region can be represented as a linear combination of the TE and TM modes. A variational Galerkin method is applied to the Helmholtz equation to develop the finite element matrix eigenvalue problem which is used to find solutions to problems with arbitrary geometries. Standard error analysis techniques are used to derive error bounds on the eigenvalues and eigenfunctions of the finite element problem. A computer program is described which will provide the user with an interactive graphical interface for the input of boundary shape and grid data, and then assemble and solve the finite element matrix equation to determine the pertinent eigenvalues and eigenfunctions representing the lower order waveguide modes. Results are then compared to published data for known waveguide geometries to demonstrate the effectiveness of the program.