In the theory of smooth dynamical systems local and global characteristics are used to describe the ''chaoticity'' of the system. In this series of lectures we want to present three of those characteristics: Entropy as a measure of combinatorial complexity, dimension as a measure of geometric complexity and Lyapunov exponents as a measure of dynamical complexity. We will consider both the case of invariant sets and invariant measures. While the theory for sets seems still far from having a satisfactory teory, the theory for invariant measures is more developed. We will explain both concepts and indicate some of the open problems. Besides the general theory we will illustrate the methods and results on examples. The second part of the course will be devoted to more special questions and recent developments. This will include standard and nonstandard thermodynamic formalism. In this area we will pose new questions and consider simple examples. Besides this general outline we will give some applications to other fields of mathematics.
