Categories, $n$-categories, bicategories, double categories, multicategories, monoidal categories, and monoids are all examples of algebraic structures on diagrams of combinatorial cells. Many of these structures have features in common with categories, such as a nerve functor, a theory of enrichment, a notion of (co)limits, or a version of the Yoneda lemma. We begin here a program to unify these and other common features into general constructions to form a shape independent category theory" that can apply to a wide variety of algebraic higher" category structures. We start with a technical treatment of ``familial monads" on presheaf categories, where each of the structures above form the category of algebras of such a monad. Using a relationship between familial functors and polynomial diagrams in $\Cat$, we establish an equivalence between familial monads and the combinatorial data of how arrangements of cells are composed in an algebraic higher category. This data provides a language for describing different types of higher categories, which we use to describe existing results on nerves of familial monad algebras and discuss the algebraic nature of their underlying cell shapes. We also construct new examples of familial monads with a focus on cubical cells. Finally, we build a theory of enrichment for any type of higher category with top-dimensional cell shapes. This shape independent construction generalizes many existing forms of enrichment, and produces new types of higher categories. The theory relies on a generalization of the wreath product of categories, which provides simple definitions of various universal constructions on categories and an elegant description of the cells in the nerve of an enriched higher category.