The first part the course will give a basic and rather self-contained introduction to the theory of Linear Feedback Shift Registers (LFSRs) as well as several applications to symmetric key cryptography and stream ciphers. In the second part of the course, the basic properties of nonlinear shift registers will be analyzed. In fact, many practical cryptosystems contain LFSRs combined in nonlinear ways. So, we will study stream ciphers based on clock controlled generators where typically one shift register controls the clocking of another shift register. In the last part of the course, we will give examples of stream ciphers and most basic cryptanalytic attacks such as correlation attacks and fast correlation attacks. If time permits, we will give some relations between sequences constructed by LFSRs and other topics in cryptography such as linear cryptanalysis, differential cryptanalysis, authentication codes, Boolean functions, bent functions etc.