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Previous model order reduction methods fit into the framework of identifying the low-order linear subspace and using the linear projection to project the full state space into the low-order subspace. Despite its simplicity, the macromodel might automatically include redundancies. In this paper, we present a model order reduction approach, named maniMOR, which extends the linear projection framework to a general nonlinear projection framework. The two key ideas of maniMOR are (1) it explicitly separates the construction of the low-order subspace and projection operation; (2) it constructs a nonlinear manifold which captures important system responses and defines the corresponding nonlinear projection operator. The low-order manifold subspace in maniMOR is identified by stitching together the low-order linear subspaces around a set of sample points on the manifold. After the manifold is determined, it is embedded into a global nonlinear coordinate system. The projection function is defined in a piece-wise linear manner, and the model evaluation is conducted directly in the manifold subspace using cheap matrix-vector product computations. As a result, a compact model is generated by precomputing all the functions and Jacobians and storing them in a look-up table. We apply maniMOR on two analog circuits and a bio-chemical system to validate its correctness. Extensive comparisons with the results of the full model and other macromodels are provided. Experimental results show that maniMOR manages to obtain a huge reduction -- e.g., from 52 to 5 for the I/O buffer circuit and from 304 to 30 for yeast pheromone pathway system. This is less than half of the size of the TPWL model with the same accuracy. With great promise to capture important system responses, maniMOR presents a novel and powerful paradigm for nonlinear model reduction, and casts inspirations for further researches.