Abstract: Diagram algebras appear in many contexts, for example braids, representation theory and statistical mechanics. There are varied examples from something as simple as a symmetric group to various partition algebras that are still emerging. We explore some examples of diagram algebras, and examine what common properties are interesting to think about. We define the concept of a cellular algebra, relate it to diagram algebras, and look at consequences for representation theory. We also look at the concept of iterated inflation and building new algebras from old.