Abstract: A central problem in approximation theory is to compute the dimension of the space of piecewise polynomial functions (splines) on a partition of a domain inside of an n-dimensional real vector space. We will describe how this can be translated into an algebraic problem, allowing use of techniques from commutative algebra and algebraic geometry. The flexibility of algebraic techniques allows a uniform approach to computing these dimension formulas. We will survey some of the results that have been obtained in this way, as well as difficulties that still persist in spite of the efforts of many researchers. We will focus primarily on planar examples, and no knowledge of the objects involved will be assumed.