In the present work, we simulate the swimming of animal species such as lampreys, eels and nematodes which are naturally modelled as a one-dimensional flexible rod (see, for example, Fauci (1996) and Fauci and Peskin (1988)). Computations are performed with the Reissner-Simo model for the dynamics of beams, which is an extension of the Kirchhoff-Love model. The rod is immersed in a three-dimensional expanse of a viscous Newtonian fluid and fluid-structure interaction is handled using the immersed boundary method of Peskin (2002). We begin our presentation of the numerical results with a study of the computed three-dimensional dynamics of both the closed and open rod model under a variety of different conditions of twist and intrinsic curvature. These are verified by comparison with the results of Lim et al. (2008, 2010). We then proceed to represent the body of swimmers in two different Reynolds number regimes using the rod model. The first (at low Reynolds number ($Re < 1$)) is relevant to the swimming cycle of wildtype C. elegans and comparisons are made with some experimental data from the research group of Arratia (www.seas.upenn.edu/∼parratia) based upon their curvature data for three swimming cycles in a buffer solution of viscosity equal to 1mPa.s. The second set of simulations is performed at much higher Reynolds numbers ($O(10^4$)) and the computed results are compared with those for an eel-like robot by Boyer et al. (2008).