It is known that the solution of regularization and interpolation problems with Hilbertian penalties can be expressed as a linear combination of the data. This very useful property, called the representer theorem, has been widely studied and applied to machine learning problems. Analogous optimality conditions have appeared in other contexts, notably in matrix regularization. In this paper we propose a unified view, which generalizes the concept of representer theorems and extends necessary and sufficient conditions for such theorems to hold. Our main result shows a close connection between representer theorems and certain classes of regularization penalties, which we call orthomonotonefunctions. This result not only subsumes previous representer the-orems as special cases but also yields a new class of optimality conditions, which goes beyond the classical linear combination of the data. Moreover, orthomonotonicity provides a useful criterion for testing whether a representer theorem holds for a specific regularization problem.