This talk will be a survey of the use of algebraic and combinatorial methods to study certain sets of piecewise polynomial functions (also known as splines) defined on a triangulated region of Rⁿ. These functions are used extensively throughout applied mathematics and engineering, but they lend themselves to an interesting theoretical study as well. For example, the sets can be viewed as subrings of the Stanley-Reisner face ring of a simplicial complex.