Advanced Differential Equations provides unique coverage of high-level topics in ordinary differential equations and dynamical systems. The textbook deliberately presents difficult material in an accessible manner by utilizing easier, friendlier notations and multiple examples. Divided into three parts, this valuable resource first focuses on standard topics such as existence and uniqueness for scalar and systems of differential equations, motivating the topics with examples and easing into the underlying theory. In the second section, the textbook focuses on the dynamic of the systems, addressing stability thoroughly; again motivating the topics with interesting examples, examining the eigenvalues of an accompanying linear matrix, as well as covering the existing literature on the topic. From a discussion of the limitations of the eigenvalues' approach, the text expands coverage the Lyapunov direct method, to support the study of stable and unstable manifolds and bifurcations. The third part of the book is devoted to the study of delay differential equations, extending from ordinary differential equations through the extension of Lyapunov functions to Lyapunov functionals. In this final section, the text explores fixed point theory, neutral differential equations, and neutral Volterra integro-differential equations. Requiring only minimal familiarity with real analysis and differential equations, Advanced Differential Equations provides students and professors with the background and practice through comprehensive exercises, such that, by the completion of Chapter 9, readers will be ready to conduct meaningful research in delay differential systems.