The goal of this paper is to present an overview of the theory of conceptual fields, illustrated by the examples of additive structures and multiplicative structures. The main difficulties encountered by students in learning mathematics and other scientific topics are conceptual. These difficulties cannot be analysed if student´s conceptions are investigated independently of the schemes that organize their behavior in problem solving situations, and if a concept or a class of situations is examined in isolation. A conceptual field can be defined as both a set of classes of situations, and as a set of interconnected concepts. This paper presents a theoretical analysis of the relationship between explicit knowledge and the implicit operational invariants that underlie schemes. Examples of theorems-in-action and of the relationships between concepts and symbolic representations are presented, as well as ways of using the framework of conceptual fields to devise intelligent computer tools.