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The present book provides a unified and general framework for studying quantum and classical dynamical systems, both finite and infinite, conservative and dissipative. Special attention is paid to the use of statistical and geometrical techniques, such as multitime correlation functions, quantum dynamical entropy, and non-commutative Lyapunov exponents, for systems with a complex evolution. The material is presented in a concise but self-contained and mathematically friendly way. The main ideas are introduced and illustrated by numerous examples which are directly connected to the relevant physics. Suggestions for further reading are included at the end of each chapter. The book addresses graduate students both in physics and mathematics with interests in mathematical aspects of quantum physics and applications of ergodic theory, operator algebras and statistics to physics, but without any prior knowledge of these subjects.