This paper presents an adaptive flux observer to estimate stator flux linkage and stator inductances of an interior permanent-magnet synchronous machine considering magnetic saturation. The concept of static and dynamic inductances due to saturation is introduced in the machine model to describe the relationship between current and flux linkage and the relationship between their time derivatives. A flux observer designed in the stationary reference frame with constant inductance is analyzed in the rotor reference frame by a frequency-response characteristic. An adaptive algorithm for an on-line inductance estimation is proposed and a Lyapunov-based analysis is given to discuss its stability. The dynamic inductances are estimated by using Taylor approximation based on the static inductances estimated by the adaptive method. The simulation and experimental results show the feasibility and performance of the proposed technique.