This book reports on a set of advances relating to nonlinear observer design, with a special emphasis on high-gain observers. First, it covers the design of filters and their addition to the observer for reducing noise, a topic that has been so far neglected in the literature. Further, it describes the adaptive re-design of nonlinear observers to reduce the effect of parametric uncertainty. It discusses several limitations of classical methods, presenting a set of successfull solutions, which are mathematically formalised through Lyapunov stability analysis, and in turn validated via numerical simulations. In the second part of the book, two applications of the adaptive nonlinear observers are described, such in the estimation of the liquid water in a hydrogen fuel cell and in the solution of a common cybersecurity problem, i.e. false data injection attacks in DC microgrids. All in all, this book offers a comprehensive report on the state-of-the-art in nonlinear observer design for energy systems, including mathematical demonstrations, and numerical and and experimental validations.