In this work, waveguides with simultaneous negative dielectric permittivity and magnetic permeability, otherwise known as left-handed waveguides, are investigated. An approach of formulating and solving an eigenvalue problem with finite element method resulting in the dispersion relation of the waveguides is adopted in the analysis. Detailed methodology of one-dimensional scalar and two-dimensional vector finite element formulation for the analysis of grounded slab and arbitrary shaped waveguides is presented. Based on the analysis, for waveguides with conventional media, excellent agreement of results is observed between the finite element approach and the traditional approach. The method is then applied to analyze left-handed waveguides and anomalous dispersion of modes is found. The discontinuity structure of a left-handed waveguide sandwiched between two conventional dielectric slab waveguides is analyzed using mode matching technique and the results are discussed based on the inherent nature of the materials. The scattering characteristics of a parallel plate waveguide partially filled with left-handed and conventional media are also analyzed using finite element method with eigenfunction expansion technique.