In this talk, I will describe recent work in string phenomenology from the perspective of computational algebraic geometry. I will begin by reviewing some of the long-standing issues in heterotic model building and the goal of producing realistic particle physics from string theory. This goal can be approached by creating a large class of heterotic models which can be algorithmically scanned for physical suitability. I will outline a well-defined set of heterotic compactifications over complete intersection Calabi-Yau manifolds using the monad construction of vector bundles. I will also describe how a combination of analytic methods and computer algebra can provide efficient techniques for proving stability and calculating particle spectra.