Contrary to traditional formulations of rewriting, which tend to rely on ideas such as category-theoretic double pushout constructions, the novel rule algebra framework is based upon the idea of interpreting rewriting rules as generators of certain diagrammatic algebras. In this whiteboard talk, I will present the core mathematical ideas and results, including the deep link between certain types of combinatorial Hopf algebras and rewriting theories. Some illustrative examples for the cases of discrete and graph rewriting will be provided.