The turbulent transport of a passive scalar (Corrsin's problem with diffusion) and of an active scalar in stably stratified fluids is studied using linear analysis (rapid distortion theory or RDT), Kraichnan's direct interaction approximation (DIA) and direct numerical simulation (DNS). The results are compared with each other and with laboratory experiments. The numerical results compare favorably with the experiments of Sirivat and Warhaft, Budwig, Tavoularis and Corrsin and Stillinger and Itsweire of van Atta's group. The RDT study reveals that much of the qualitative behavior observed in experiments, such as the tendency for the system to evolve towards some statistically asymptotic state, is embodied in the linear results. Specifically, predictions from the passive scalar linear theory for lengthscale ratios (both integral and microscale) are in good agreement with nonlinear results but the linear values for the scalar transport correlation coefficient contain significant error;DNS and DIA results are reasonably close for the velocity field and scalar transport with Gaussian initial spectra, but differ significantly for exponential spectra. Although the transport problem is anisotropic, the DIA results using only the first Legendre functions yield integrated results identical to those obtained with two harmonics. Linear and DIA runs show significant dependance of scalar transport upon initial spectral shapes;The passive scalar transport problem is shown to be equivalent to the sum of an isotropic scalar turbulence problem with the scalar initial conditions of the transport problem and a transport problem with zero initial scalar fluctuations (both with the same velocity fields). The asymptotic state is defined by the zero initial scalar field transport problem while the rate and manner in which the complete problem approaches this state is strongly affected by the initial scalar field;The DIA is also used to simulate some experimental decaying turbulence experiments in isotropic velocity fields, passive isotropic scalar fields and transport of a passive scalar by an isotropic velocity field. A rational technique for determining the appropriate nondimensionalization for time is presented and demonstrated. The spectral aspect ratio, the ratio of the integral lengthscale and Taylor microscale, A, is shown, with R[subscript][lambda], to be important for accurate simulation of the evolution of the isotropic velocity field. For problems involving a scalar field, the ratio of this number for the velocity and scalar fields plays a crucial role in the subsequent evolution of the scalar field.