Phase transition dynamics are central to nonlinear sciences, encompassing both classical and general transitions. This book introduces a comprehensive dynamic transition theory for dissipative systems and its applications across various scientific fields. The theory classifies dynamic transitions into three categories: continuous, catastrophic, and random. Continuous transitions involve gradual changes near the basic state, catastrophic transitions involve abrupt changes, and random transitions involve stochastic behavior. The theory is grounded in the search for a system's full set of transition states, represented by local attractors, and is motivated by phase transition problems in nonlinear sciences. The book combines modeling, mathematical analysis, and physical predictions to explore equilibrium and nonequilibrium phase transitions. It addresses applications in statistical physics, fluid dynamics, climate dynamics, chemistry, and biology, including examples such as Rayleigh-Bénard convection, El Niño-Southern Oscillation, and Belousov-Zhabotinsky reactions. The theory provides a systematic approach to understanding the types and structures of transitions, with applications to a wide range of scientific problems. The book is intended for applied mathematicians, physicists, and researchers in related fields, offering a detailed exploration of dynamic transitions and their implications.Phase transition dynamics are central to nonlinear sciences, encompassing both classical and general transitions. This book introduces a comprehensive dynamic transition theory for dissipative systems and its applications across various scientific fields. The theory classifies dynamic transitions into three categories: continuous, catastrophic, and random. Continuous transitions involve gradual changes near the basic state, catastrophic transitions involve abrupt changes, and random transitions involve stochastic behavior. The theory is grounded in the search for a system's full set of transition states, represented by local attractors, and is motivated by phase transition problems in nonlinear sciences. The book combines modeling, mathematical analysis, and physical predictions to explore equilibrium and nonequilibrium phase transitions. It addresses applications in statistical physics, fluid dynamics, climate dynamics, chemistry, and biology, including examples such as Rayleigh-Bénard convection, El Niño-Southern Oscillation, and Belousov-Zhabotinsky reactions. The theory provides a systematic approach to understanding the types and structures of transitions, with applications to a wide range of scientific problems. The book is intended for applied mathematicians, physicists, and researchers in related fields, offering a detailed exploration of dynamic transitions and their implications.