In this work, three normalized parameters are used to study the dispersion characteristics of three different asymmetric slab waveguide structures. Each structure contains at least one layer made of left-handed material (LHM). The first structure comprises a LHM cladding layer, the second contains a LHM substrate layer, and the third has a LHM cladding and substrate. The results show that the dispersion characteristics of these structures differ from conventional guided waves in waveguide structures comprising right-handed materials. A number of unusual properties are observed including the dispersion characteristics of the fundamental mode for the first structure is different for different μ c and μ s . The fundamental mode exists and can only propagate within a small frequency range in the second structure. In the third structure, the fundamental mode does not exist whereas the first-order mode exists in a narrow frequency range. No propagation modes in some frequency ranges is observed which enables the structure to behave as a bandpass filter.