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In this study, differential stability is introduced for non-linear systems, and this concept is further exploited in reduced-order observer design for non-linear systems with non-linearities of unmeasured state variables, a more general class of non-linear systems than the systems with linear observer errors. It has been shown that if the dynamics of unmeasured state variables under a state transformation is differentially stable, a reduced-order observer can be designed to produce asymptotically convergent estimates of the unmeasured state variables. A systematic design method is then introduced for a class of multi-output non-linear systems. For such a system, a non-linear term of the unmeasured state variables enter the system through a coupling matrix. It is found that a reduced-order observer can be designed if the linear part with the coupling matrix as the input matrix has no unstable invariant zeros. A further exploitation is presented for a class of single-output non-linear systems with non-linearity of unmeasured state variables. In this case, the coupling vector is allowed to be a vector field which depends on the system output.