A family of connections on the space of couplings for a renormalizable field theory is defined. The connections are obtained from a Levi-Civita connection, for a metric which is a generalisation of the Zamolodchikov metric in two dimensions, by adding a family of tensors which are solutions of the renormalization group equation for the operator product expansion co-efficients. The connections are torsion free, but not metric compatible in general. The renormalization group flows of N = 2 supersymmetric Yang-Mills theory in four dimensions and the O(N)-model in three dimensions, in the large N limit, are analysed in terms of parallel transport under these connections. Â© 1999 Published by Elsevier Science B.V. All rights reserved.