Abstract: We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of learning in terms of various combinatorial quantities. In the setting of equivalence plus membership queries, we give an algorithm which learns a class in polynomially many queries whenever any such algorithm exists. Our approach is flexible and powerful enough to enough to give new and very short proofs of the efficient learnability of several prominent examples (e.g. regular languages and regular omega-languages), in some cases also producing new bounds on the number of queries.