In multidimensional separations, two or more independent separation methods are coupled in an effort to resolve complex mixtures. The displacement mechanisms of each method should be orthogonal, such that little correlation exists between the retention of compounds in each dimension. When multiple orthogonal separation methods are coupled such that all sample components are subjected to complete analysis on all dimensions, the method is considered "comprehensive". The primary advantage of comprehensive multidimensional separations over their one-dimensional counterparts is the potential for dramatically enhanced resolution. High resolving power can be achieved because the peak capacity of a comprehensive multidimensional separation is roughly equal to the product of the individual peak capacities of each dimension. In this review, the theory and instrumentation of two-dimensional liquid chromatography (LC-LC) and liquid chromatography-capillary electrophoresis (LC-CE) separations are discussed. Some applications of these techniques to the separation of biological molecules are highlighted. Future directions for the development of multidimensional separations are also considered.