This book focuses on the nonlinear dynamics based on the vector fields with univariate quadratic functions. This book is a unique monograph for two—dimensional quadratic nonlinear systems. It provides different points of view about nonlinear dynamics and bifurcations of the quadratic dynamical systems. Such a two—dimensional dynamical system is one of simplest dynamical systems in nonlinear dynamics, but the local and global structures of equilibriums and flows in such two—dimensional quadratic systems help us understand other nonlinear dynamical systems, which is also a crucial step toward solving the Hilbert’s sixteenth problem. Possible singular dynamics of the two—dimensional quadratic systems are discussed in detail. The dynamics of equilibriums and one—dimensional flows in two—dimensional systems are presented. Saddle—sink and saddle—source bifurcations are discussed, and saddle—center bifurcations are presented. The infinite—equilibrium states are switching bifurcations for nonlinear systems. From the first integral manifolds, the saddle—center networks are developed, and the networks of saddles, source, and sink are also presented. This book serves as a reference book on dynamical systems and control for researchers, students, and engineering in mathematics, mechanical, and electrical engineering.