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The propagation characteristics of twisted hollow waveguides are considered, and various analysis methods are proposed. It is shown that a twisted hollow waveguide can support waves that travel at a speed slower than the speed of light c. These modes are of particular interest, as slow wave structures have many potential applications in accelerators and electron traveling wave tubes. Since there is no exact closed form solution for the electromagnetic fields within a twisted waveguide or cavity, several previously proposed approximate methods are examined. It is found that the existing perturbation theory methods yield adequate results for slowly twisted structures; however, our efforts here are geared toward analyzing rapidly twisted structures using newly developed finite-difference methods. To validate the results of the theory and simulations, rapidly twisted cavity prototypes have been designed, fabricated, and measured. These measurement results are compared to simulated results, and very good agreement has been demonstrated.