This book explores the properties of evolving systems, necessitating the advancement of existing models of Dynamic Systems for formal representation and study. It introduces a new class of dynamic models based on nonlinear integral-functional equations with a historical context. Theorems on the existence and uniqueness of solutions are proven, and both qualitative and numerical investigations of the solution properties are conducted. The book addresses various optimization problems, examines the modeling of resource distribution between macroeconomic sectors A and B, investigates mechanisms regulating the redistribution of cells in immune and hematopoietic tissues, and models the process of sugar accumulation in sugar beets, among other challenges. A range of tools for simulating evolving systems is also discussed.