This book summarizes and highlights progress in Dynamical Systems achieved during six years of the German Priority Research Program "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," funded by the Deutsche Forschungsgemeinschaft (DFG). The three fundamental topics of large time behavior, dimension, and measure are tackled with by a rich circle of uncompromisingly rigorous mathematical concepts. The range of applied issues comprises such diverse areas as crystallization and dendrite growth, the dynamo effect, efficient simulation of biomolecules, fluid dynamics and reacting flows, mechanical problems involving friction, population biology, the spread of infectious diseases, and quantum chaos. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications far into the neighboring disciplines of science.