It is common practice in algebraic topology to study spaces by analogy with simpler combinatorial objects. By far the most prevalent are simplicial sets, which describe spaces as being built out of simplices in each dimension and have a homotopy theory equivalent to that of topological spaces. In a similar fashion, cubical sets describe spaces as built out of cubes in each dimension. Globular sets allow spaces to be built by specifying points, paths between those points, 2-dimensional paths between those paths, and so on in each subsequent dimension. Each of these objects is defined as a contravariant functor from some category of “cells” (like simplices or cubes) into sets, and each can be used to describe higher homotopy in topology, higher morphisms in category theory, and higher equalities in type theory. I will show how different cell shapes have distinct advantages in modeling this higher order information, with particular focus on how each encodes the algebraic structure corresponding to path concatenation in homotopy theory, composition in category theory, and transitivity of equality in type theory.