Book excerpt: Adaptive control is the control method used by a controller which adapt to a system with unknown or varying parameters. As a newly developed technique, L1 adaptive control has drawn increased attention in past decades. The key feature of L1 adaptive control architecture is guaranteed robustness in the presence of fast adaptation. With L1 adaptive control architecture, fast adaptation appears to be beneficial both for performance and robustness, while the trade-off between the two is resolved via the selection of the underlying filtering structure. The latter can be addressed via conventional methods from classical and robust control. Moreover, the performance bounds of L1 adaptive control architectures can be analyzed to determine the extent of the modeling of the system that is required for the given set of hardware. The main contribution of this dissertation is to extend the framework of L1 adaptive control theory and applied in various of applications. It can be summarized with 3 different parts: The first one is the extension of L1 adaptive to time-varying system and non-minimum phase system by using eigenvalue assignment method. This approach has been demonstrated by both theoretical models as well as high fidelity models such as flexible wing aircraft model from NASA and also the supersonic glider model developed from the supersonic lab in Austria. The 2nd part focuses on filter bandwidth adaptation in the L1 adaptive control architecture. The stability condition of the low-pass filter in control is relaxed by introducing an additional Lyapunov-based adaptation mechanism which results in a more systematic design with minimized tuning efforts. Adaptability for arbitrarily large nonlinear time-varying uncertainties without redesign parameters. The overall system is a non-LTI design even in the limiting case. The 3rd part introduces the concept of predictive horizon and online optimization into L1 adaptive control. This approach enables L1 adaptive control to solve the output limitation even for the non-minimum phase system.