This volume presents a comprehensive introduction into rigorous geometrical dynamics of complex systems of various natures. By "complex systems", in this book are meant high-dimensional nonlinear systems, which can be (but not necessarily are) adaptive. This monograph proposes a unified geometrical approach to dynamics of complex systems of various kinds: engineering, physical, biophysical, psychophysical, sociophysical, econophysical, etc. As their names suggest, all these multi-input multi-output (MIMO) systems have something in common: the underlying physics. Using sophisticated machinery composed of differential geometry, topology and path integrals, this book proposes a unified approach to complex dynamics – of predictive power much greater than the currently popular "soft-science" approach to complex systems. The main objective of this book is to show that high-dimensional nonlinear systems in "real life" can be modeled and analyzed using rigorous mathematics, which enables their complete predictability and controllability, as if they were linear systems. The book has two chapters and an appendix. The first chapter develops the geometrical machinery in both an intuitive and rigorous manner. The second chapter applies this geometrical machinery to a number of examples of complex systems, including mechanical, physical, control, biomechanical, robotic, neurodynamical and psycho-social-economical systems. The appendix gives all the necessary background for comprehensive reading of this book.