It has been over 50 years since Toms [B.A. Toms, Proc. 1^st Int. Congr. Rheol., Sec. II, 135, North Holland (1949)] discovered that adding small quantities of long-chain polymer to turbulent pipe flow could drastically reduce the amount of turbulent drag. Since then a substantial amount of research has gone into examining dilute polymer solutions and their turbulent drag reducing properties, yet the precise mechanism by which the polymers interact with the turbulence to produce this affect is still unclear. From a theoretical standpoint, the difficulty lies in the analysis of accurate models of the fluid dynamics. The combination of the complex mathematical description required for turbulence and the intricate constitutive equations of polymer motion poses significant problems. However, recent developments in computing have resulted in machines powerful enough to simulate such flows using direct numerical simulation (DNS); a technique whereby all the important scales of turbulent fluid motion are fully resolved using an algorithm derived from the full momentum conservation equations. DNS has been successfully employed in previous studies of dilute polymer solutions in computational domains similar to experimental apparatus such as pipe flow [J. M. J. den Toonder, M. A. Hulsen, G. D. C. Kuiken and F. T. M. Nieuwstadt, J. Fluid Mech., 337, 193 (1997)] and channel flow, [R. Sureshkumar, A. N. Beris and R. A. Handler, Phys. Fluids, 9(3), 743 (1996)]. It was our aim to identify the changes within the turbulent dynamics produced by the presence of polymers in homogeneous isotropic turbulence - without an dependence on solid boundary conditions. We decided to simulate our solutions within a infinitely repeating cube subject to periodic boundary conditions, a well established technique for Newtonian fluids. In this thesis we examine a range of spectral measures and integral parameters for various non-Newtonian fluid models in statistically stationary homogeneous isotropic turbulence and compare them to an equivalent Newtonian flow using DNS. We begin by outlining the general theory of homogeneous isotropic non-Newtonian turbulence in the Fourier space domain. We then demonstrate how the general non-Newtonian momentum conservation equations are adapted for use in our DNS. This is followed by a literature review of turbulent drag reduction by long-chain polymer additives. The remainder of the thesis is concerned with the new work. We outline the four non-Newtonian fluid models we applied and present the results obtained from the DNS calculations. Each model embodies a particular characteristic of polymer solutions. The first is of our own construction and is based on the ability of polymers to increase the viscosity of the solution at small scales. Second, we simulate the nonlinear model of McComb [W. D. McComb, Int. J. Engng. Sci., 14, 239 (1976)] where the stress exhibits a nonlinear dependence on the rate of strain. For both of these we were able to obtain an analytical expression for the energy spectra and compared these to the DNS results. The third model is the viscous anisotropic model of den Toonder et al. (see above reference) which introduces a directional element based on the orientation of the polymers in the flow, by assuming they are of constant length and align with the instantaneous velocity. Finally we model a fully coupled FENE-P fluid [L. E. Wedgwood and R. B. Bird, Ind. Eng. Chem. Res., 27, 1313 (1988)] in which the polymers are finitely extensible, elastic and have their own equation of motion giving their orientation. In this way we have identified changes within the structure of turbulence itself which may be related to the drag reduction phenomenon.