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The problem of constructing globally convergent, reduced-order observers for general nonlinear systems is addressed. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected (invariant) manifold in the extended state space. Current results on nonlinear observer design require that the nonlinearities appearing in the system equations are either linear functions of the unmeasured states or monotonic functions of a linear combination of the states. In this paper we relax these two assumptions by allowing for a wider class of nonlinearities to appear in the system equations. The proposed approach is applied on several examples including a perspective vision system and a general two-degrees-of-freedom mechanical system.