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One of the biggest discoveries in the theory of dynamical systems was that smooth (deterministic) systems can behave very randomly. Since then a rich theory of chaotic properties of smooth dynamical systems was developed using geometric, topological and probabilistic methods. In the talk we will present ideas, highlight main results and discuss techniques that were developed during the last 70 years. In the second part we plan to discuss more recent advancements and present main open questions in the field.