Investigations on control and optimization of continuum manipulators have resulted in a number of kinematic and dynamic modeling approaches each having their own advantages and limitations in various applications. In this paper, a comparative study of five main methods in the literature for kinematic, static and dynamic modeling of continuum manipulators is presented in a unified mathematical framework. The five widely used methods of Lumped system dynamic model, Constant curvature, two-step modified constant curvature, variable curvature Cosserat rod and beam theory approach, and series solution identification are re-viewed here with derivation details in order to clarify their methodological differences. A comparison between computer simulations and experimental results using a STIFF-FLOP continuum manipulator is presented to study the advantages of each modeling method.