This thesis is concerned with the optimal flight trajectories in the presence of a three-dimensional windshear. Both the take-off and abort landing problems are studied. A mathematical model of a three-dimensional windshear is developed by the superposition of the flow fields of two symmetric vortex rings with appropriate parameters (circulation strength, radius, height). The flow field produced by this vortex ring pair is close to that of a real microburst. The wind components are functions of the geometric coordinates and can be obtained using either the Biot-Savart law or the properties of the stream function. With this wind model, the strongest windshear and downdraft are located in a vertical plane passing through the central axis of the vortex ring pair. Therefore, in the computation of flight trajectories, the aircraft is assumed to fly in this vertical plane. Two cases are considered: (A) at the initial time, the aircraft is located in the region of strongest headwind; (B) at the initial time, the aircraft is located in the region of weak-to-moderate headwind. Case A implies late detection, while Case B implies early detection of windshear. Optimal trajectories are computed for both take-off and abort landing. For the take-off problem, the performance index being minimized is the peak value of the deviation of the absolute path inclination from a reference value; for the abort landing problem, the performance index being minimized is the peak value of the altitude drop. The resulting optimal control problems are Chebyshev problems, which are converted into Bolza problems via suitable transformations. Then, the Bolza problems are solved by using the sequential gradient-restoration algorithm (SGRA). Numerical computations for both the take-off and abort landing problems lead to the following conclusions: (i) The survival capability of the optimal trajectory is superior to that of the constant pitch trajectory and the maximum angle of attack trajectory; this means that near-optimal guidance schemes should be developed to improve the survival capability of an aircraft in a severe windshear. (ii) For the optimal trajectories, the survival capability in Case B (early detection) is superior to that in Case A (late detection); this indicates that early detection of a windshear can enhance the safety of flight.