Properties of derivative expansion approximations to the renormalization group

Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group into a set of partial differential equations which at fixed points become nonlinear eigenvalue equations for the anomalous scaling dimension eta. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the role of reparametrization invariance, and the convergence of the derivative expansion is addressed.