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A compact nonlinear model order-reduction method (NORM) is presented that is applicable for time-invariant and periodically time-varying weakly nonlinear systems. NORM is suitable for model order reduction of a class of weakly nonlinear systems that can be well characterized by low-order Volterra functional series. The automatically extracted macromodels capture not only the first-order (linear) system properties, but also the important second-order effects of interest that cannot be neglected for a broad range of applications. Unlike the existing projection-based reduction methods for weakly nonlinear systems, NORM begins with the general matrix-form Volterra nonlinear transfer functions to derive a set of minimum Krylov subspaces for order reduction. Moment matching of the nonlinear transfer functions by projection of the original system onto this set of minimum Krylov subspaces leads to a significant reduction of model size. As we will demonstrate as part of comparison with existing methods, the efficacy of model reduction for weakly nonlinear systems is determined by the achievable model compactness. Our results further indicate that a multipoint version of NORM can substantially improve the model compactness for nonlinear system reduction. Furthermore, we show that the structure of the nonlinear system can be exploited to simplify the reduced model in practice, which is particularly effective for circuits with sharp frequency selectivity. We demonstrate the practical utility of NORM and its extension for macromodeling weakly nonlinear RF communication circuits with periodically time-varying behavior.