The theory of congruence lattices of universal algebras is one of the most rich and developed parts of contemporary algebra. Unfortunately, the rather special and purely internal definition of the congruence relation does not extend directly to structures other than sets with operations; in fact it even does not apply to the slightly more general models (sets with operations and relations). A way out is to simply omit the relations; however, this is not quite satisfactory: for example, take the linearly ordered additive group \(\mathbb{Z}\) of integers and note that the quotient modulo a non-trivial congruence is a finite group \(\mathbb{Z}_ n\) which cannot be linearly ordered. The same difficulties typically occur in non-algebraic structures, e.g., what is a congruence on an ordered set, graph or topological space? A certain solution is provided by categories, where congruences are defined via kernel pairs of morphisms. Of course, this definition is external and relative because the congruences depend not only on the structure but also on the choice of the category. This paper presents a compromise between the general, categorial approach and the quite special universal algebraic one.