In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of "category"; for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.