A new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced-order bases is presented. It is particularly suited for problems characterized by different physical regimes, parameter variations, or moving features such as discontinuities and fronts. Instead of approximating the solution of interest in a fixed lower-dimensional subspace of global basis vectors, the proposed model order reduction method approximates this solution in a lower-dimensional subspace generated by most appropriate local basis vectors. To this effect, the solution space is partitioned into subregions, and a local reduced-order basis is constructed and assigned to each subregion offline. During the incremental solution online of the reduced problem, a local basis is chosen according to the subregion of the solution space where the current high-dimensional solution lies. This is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower-dimensional solution space. Because it is also applicable to the process of hyper reduction, the proposed method for nonlinear model order reduction is computationally efficient. Its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid-structure-electric interaction problems. Copyright © 2012 John Wiley & Sons, Ltd.