This article develops a unified framework to design fuzzy-model-based observers of general nonlinear systems for both discrete-time and continuous-time cases. This observer problem is known as a challenging task due to the mismatch caused by the unmeasurable premise variables. To deal with this major challenge, we propose to rewrite the nonlinear system as a specific fuzzy model with two types of local nonlinearities: measurable and unmeasurable. Then, a differential mean value theorem for vector-valued functions is applied to local unmeasurable nonlinearities. This allows to represent the estimation error dynamics in a special polytopic form involving measurable membership functions and unknown but bounded time-varying parameters. Using Lyapunov-based arguments, design conditions in terms of linear matrix inequalities are derived to guarantee the asymptotic convergence of the estimation error. Three illustrative examples are given to demonstrate the interests of the new fuzzy observer framework in reducing: 1) the design conservatism and 2) the numerical complexity of the fuzzy observer structure for real-world applications.