Earlier chapters deal with abductive inferences to explanations which are deductive or inductive-probabilistic. This more or less standard account has so far ignored the fact that explanatory and predictive success in science is often approximate. Therefore, the analysis of abduction should cover also approximate explanations, which is illustrated by Newton’s explanation of Kepler’s harmonic law (Sect. 8.1). The notions of approximate truth (closeness to being true), verisimilitude (closeness to complete qualitative or quantitative truth) and legisimilitude (closeness to the true law) are defined in Sect. 8.2. This leads us to generalize Peirce’s model of abduction to cases where the conclusion states that the best theory is truthlike or approximately true, with illustrations from idealized theories and models (Sect. 8.3). In a comparative formulation, if theory Y is a better explanation of the available evidence E than theory X, then conclude for the time being that Y is more truthlike than X. To justify such abduction, we need a method of estimating degrees of truthlikeness by their expected values. Another tool is the notion of probable approximate truth. Then, in order to answer to Laudan’s challenge, the probabilistic link between empirical success and truth has to be replaced with a fallible bridge from the approximate empirical success of a theory to its truthlikeness (Sect. 8.4). Section 8.5 gives some remarks on abductive belief revision, which is related to cases where the evidence is conflict with the theory. This theme extends Aliseda’s way of linking belief revision models with abductive reasoning.