The monograph deals with a number of fundamental issues related to nonlinear dynamics and chaos. In particular, new definitions of an invariant chaotic set of a dynamical system and a chaotic attractor are given. The definitions proposed here allow us to detect a new type of chaotic behavior, which is realized in an uncompact and infinite-dimensional case - the so-called turbulent chaos. The content of this phenomenon is illustrated by a specific example that allows for rigorous mathematical analysis. Other topics covered in this book include the mathematical aspects of Landau's theory of turbulence. Namely, the feasibility of the Landau scenario in its generalized version is illustrated by a number of specific examples from various fields of natural science. Some other typical situations are also studied when, when changing the control parameter in the system, a chaotic attractor arises or there are quite a few different chaotic attractors (chaotic buffering). For example, a new way of accounting for rare catastrophic events in systems with complex behavior is proposed, as well as a new approach to the design of chaotic oscillation generators. For senior students, graduate students of mathematical and physical faculties of universities, specialists in applied mathematics, oscillation theory, nonlinear dynamics and chaos.