Aircraft tracking, intent inference, and trajectory predictions are important tools for enhanced capacity in air traffic control operations. In this paper, we propose an algorithm that performs these three tasks accurately. The algorithm uses a hybrid estimation algorithm to estimate the aircraft's state and flight mode. These estimates are combined with knowledge about air traffic control regulations, the aircraft's flight plan, and the environment to infer the pilot's intent. Trajectory predictions are computed as a function of the aircraft's motion (state and mode estimates) and the inferred intent The result is an algorithm that provides, in real-time, accurate intent and trajectory predictions for aircraft. We analyze and test the performance of the proposed algorithm with various scenarios representative of current and future aircraft operations within the National Airspace System.