A full-vectorial contour integral equation analysis of the natural modes of dielectric waveguides (DW) of arbitrary cross section is presented. The Galerkin method, together with the Analytical Regularization procedure, is applied to discretizing and solving the eigenvalue problem. This ensures the fast convergence and superior accuracy of the numerical algorithms. The waveguide cross section is characterized by a parametrical curve defining its contour, with a limited curvature at each point. This avoids the singularity points at corner regions and provides accurate results, even for waveguides with virtually sharp corners. Both fundamental and higher order mode propagation characteristics are studied in the bound, leaky, and complex regimes. Numerical results consistent with other theories and experimental data are presented for a wide range of practical dielectric waveguides that demonstrate the efficiency, accuracy, and versatility of the method developed. Finally, the technique is applied to model a fused fiber coupler.