We present a number of general results relating to nonlinear viscous and viscoelastic anisotropic fluids with transversely isotropic symmetry. Among the topics considered are some kinematic results, the effect of kinematic constraints, the stress response in the presence of kinematic constraints. and constitutive equations for the extra-stress. The analogues for transversely isotropic fluids of the constitutive equations of isotropic Reiner-Rivlin, Rivlin-Ericksen and second-order fluids are formulated. We give some general results on viscometric flows of transversely isotropic materials, and illustrate these by examining the case of helical flow. (C) 2004 Elsevier B.V. All rights reserved.