Modern health management approaches for gas turbine engines (GTE) aim to acquire precise information about the health state of the GTE components to optimize the maintenance decisions with respect to both the economy and safety. The task becomes more challenging for the GTE parts inaccessible to direct measurements with the available sensors of the GTE control system. This article-based thesis integrates a set of five coherent research work to address this problem. A detailed nonlinear thermodynamic model for single shaft GTEs is developed to predict the expected cycle parameters for the GTE in the healthy condition. In reality, the measured cycle parameters gradually deviate from the prediction due to performance deterioration. Physics-based performance indicators are defined based on the deviations in the measured performance parameters, compared to the respective model predictions. The indicators can effectively monitor the GTE performance deterioration in both short-term and long-term regimes. In the next step, effect of the air humidity is taken into account to enhance the GTE model, and it is shown that the enhanced model can improve the performance monitoring by reducing the uncertainties. In order to separate the effects of different fault modes, an inference-based model is developed to predict the short-term recoverable performance deterioration due to the compressor fouling under different ambient and operating conditions. For the long-term non-recoverable performance deterioration due to the degradation mechanisms in the turbine hot section, two steps are undertaken; 1) a state estimation framework is developed for nonlinear/non-Gaussian systems with non-uniform time steps to track a degradation symptom of the turbine, i.e., loss of isentropic efficiency, using the observable performance indicators, and 2) the state estimation framework is extended to multidimensional dynamical systems with stochastic inputs for simultaneous tracking of two degradation symptoms, i.e., loss of isentropic efficiency and increase of the mass flow, using the observable parameters, provided by the GTE operating system. The developed techniques and frameworks are verified and validated, using a set of three-year operating data from an industrial GTE in a power plant.