The macroscopic dynamics of strongly flocculated colloidal suspensions are of both fundamental interest and practical significance across a range of applications from soil mechanics to solid-liquid separation. Over 30 years ago, Buscall and White [Buscall, R., White, L.R., 1987. The consolidation of concentrated suspensions. Part 1. The theory of sedimentation, Journal of the Chemical Society: Faraday Transactions I 83 (3), 873-891] proposed a one-dimensional (1D) theory of sedimentation and consolidation of such suspensions, which spawned suspension characterization tools and process modeling techniques to successfully predict 1D separation behavior. However, many applications such as continuous thickening or cross-flow filtration are multidimensional in practice, involving suspension transport, deformation and separation, and so cannot be described by the 1D BW theory. Multidimensional suspension mechanics has received a little attention over the past 30 years, requiring development of a tensorial rheology of suspensions, capable of describing arbitrary loadings and simultaneous shear and compression. In this paper, we develop a multidimensional theory of the macroscopic mechanics of strongly flocculated colloidal suspensions to describe suspension transport, deformation and separation in terms of the bulk suspension velocity field and local average solids volume fraction alone. This is achieved by consistent extension of the fundamentals of BW theory to multiple dimensions, with constitutive models justified and expressed in terms of experimentally measurable material functions. Constitutive aspects which are outstanding due to lack of empirical evidence are clearly identified, pointing to potential research directions regarding the tensorial rheology of colloidal suspensions. Utility of the theory is demonstrated in an example comparison between 1D and 2D continuous gravity thickener operation. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights reserved.