The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion ofrandomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enric Contents; 1 The complexity of sets; 2 The descriptive complexity of strings; 3 Martin-Löf randomness and its variants; 4 Diagonally noncomputable functions; 5 Lowness properties and K-triviality; 6 Some advanced computability theory; 7 Randomness and betting strategies; 8 Classes of computational complexity; 9 Higher computability and randomness; Solutions to the exercises; References; Notation Index; Index;