This survey of computability theory offers the techniques and tools that computer scientists as well as mathematicians and philosophers studying the mathematical foundations of computing need to mathematically analyze computational processes and investigate the theoretical limitations of computing. Beginning with an introduction to the mathematisation of mechanical process using URM programs this textbook explains basic theory such as primitive recursive functions and predicates and sequencecoding partial recursive functions and predicates and loop programs.ampampnbspAdvanced chapters cover the Ackerman function Tarskis theorem on the nonrepresentability of truth Goedels incompleteness and Rossers incompleteness theorems two short proofs of the incompleteness theorem that are based on Lobs deliverability conditions Churchs thesis the second recursion theorem and applications a provably recursive universal function for the primitive recursive functions Oracle computations and various classes of computable functionals the Arithmetical hierarchy Turing reducibility and Turing degrees and the priority method a thorough exposition of various versions of the first recursive theorem Blums complexity Hierarchies of primitive recursive functions and a machineindependent characterisation of Cobhams feasibly computable functions.