Chain complexes were not conceived on their own, but were discovered in the course of exploring various abstract models of space. An abstract simplicial complex is a representation of a space that is both structurally rich yet requires few axioms to describe. Many of these properties are elementary to state and lead to useful computational algorithms.This chapter forges the connection between abstract simplicial complexes and the algebraic constructions of the previous chapter. Given an abstract simplicial complex, one can construct its simplicial chain complex and compute its simplicial homology as algebraic summaries of its structure.