  Higher order conformal perturbation theory is studied for theories with and
without boundaries. We identify systematically the universal quantities in the
beta function equations, and we give explicit formulae for the universal
coefficients at next-to-leading order in terms of integrated correlation
functions. As an example, we analyse the radius-dependence of the conformal
dimension of some boundary operators for the case of a single Neumann brane on
a circle, and for an intersecting brane configuration on a torus, reproducing
in both cases the expected geometrical answer.
