In algebra/linear algebra we learn how to construct actions of sets endowed with specific algebraic structures on other sets. So for example, in the theory of vector spaces we have an action of a field or a skew-field. In the theory of $R$-modules we have a ring action, and in $G$-sets a group action.
I am wondering if there exist other types of 'actions' of algebraic structures in this sense. Can anyone provide examples or suggest theoretical frames where to look into?