The controllability of viscoelastic fields is a fundamental concept that defines some essential capabilities and limitations of the resulting materials. In this paper, we study the controllability of different homogeneous flow fields of viscoelastic fluids governed by the upper convected Maxwell model. The approach is largely based on the nonlinear geometric control theory. Through the analysis of the control Lie algebra, we find the submanifolds in the state space on which the homogeneous flow fields are weakly controllable. Our approach can be generalized to more complicated systems. (C) 2007 Elsevier B.V. All rights reserved.