The talk is a brief historical account of the development of the theory
that deals with the phenomenon widely known as "deterministic
chaos" — the appearance of irregular chaotic motions in purely
deterministic dynamical systems on compact phase spaces.
The hyperbolic theory of dynamical systems provides a mathematical foundation for this
paradigm and thus serves as a basis for the theory of chaos. The hyperbolic behavior can
be interpreted in various ways and the weakest one is associated with dynamical systems
with non-zero Lyapunov exponents.
I will describe various types of hyperbolicity, outline some examples of systems with
hyperbolic behavior and discuss the still-open problem of whether chaotic dynamical
systems are generic. This genericity problem is closely related to two other important
problems in dynamics on whether systems with non-zero Lyapunov exponents exist on any
compact phase space and whether chaotic behavior can coexist with a regular (non-chaotic)
one in a robust way.