  We examine the precise connection between the exact renormalisation group
with local couplings and the renormalisation of correlation functions of
composite operators in scale-invariant theories. A geometric description of
theory space allows us to select convenient non-linear parametrisations that
serve different purposes. First, we identify normal parameters in which the
renormalisation group flows take their simplest form; normal correlators are
defined by functional differentiation with respect to these parameters. The
renormalised correlation functions are given by the continuum limit of
correlators associated to a cutoff-dependent parametrisation, which can be
related to the renormalisation group flows. The necessary linear and non-linear
counterterms in any arbitrary parametrisation arise in a natural way from a
change of coordinates. We show that, in a class of minimal subtraction schemes,
the renormalised correlators are exactly equal to normal correlators evaluated
at a finite cutoff. To illustrate the formalism and the main results, we
compare standard diagrammatic calculations in a scalar free-field theory with
the structure of the perturbative solutions to the Polchinski equation close to
the Gaussian fixed point.
