The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with non-zero Lyapunov exponents. In my talk I will describe systems with non-zero Lyapunov exponents and discuss the still-open problem of how typical these systems are. I will outline some recent results in this direction and relation between this problem and recent advances in the Pugh-Shub stable ergodicity theory. The talk will be accessible to the general mathematical audience including graduate students.