Turing machines as a basic computational model for computational complexity analysis, time and space complexity on Turing machines. Alternative models of computation, RAM and RASP machines and their relation to Turing machines in the context of complexity. Asymptotic complexity estimations, complexity classes based on time- and space-constructive functions, typical examples of complexity classes. Properties of complexity classes: importance of determinism and non-determinism in the area of computational complexity, Savitch theorem, relation between non-determinism and determinism, closure w.r.t. complement of complexity classes, proper inclusion between complexity classes. Selected advanced properties of complexity classes: Blum theorem, gap theorem. Reduction in the context of complexity and the notion of complete classes. Examples of complete problems for different complexity classes. Deeper discussion of P and NP classes with a special attention on NP-complete problems (SAT problem, etc.). Relationship between decision and optimization problems. Methods for computational solving of hard optimization problems: deterministic approaches, approximation, probabilistic algorithms. Relation between complexity and cryptography.  Deeper discussion of PSPACE complete problems, complexity of formal verification methods.