Inductive Logic Programming (ILP) is a research area which investigates the construction of first-order definite clause theories from examples and background knowledge. ILP systems have been applied successfully in a number of real-world domains. These include the learning of structure-activity rules for drug design, finite-element mesh design rules, rules for primary-secondary prediction of protein structure and fault diagnosis rules for satellites. There is a well established tradition of learning-in-the-limit results in ILP. Recently some results within Valiant's PAC-learning framework have also been demonstrated for ILP systems. In this paper it is argued that algorithms can be directly derived from the formal specifications of ILP. This provides a common basis for Inverse Resolution, Explanation-Based Learning, Abduction and Relative Least General Generalisation. A new general-purpose, efficient approach to predicate invention is demonstrated. ILP is underconstrained by its logical specification. Therefore a brief overview of extra-logical constraints used in ILP systems is given. Some present limitations and research directions for the field are identified.