Feynman categories and representation theory

We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.