Singular models occur frequently in machine learning and computational biology. In this talk, we give a basic introduction to Sumio Watanabe\'s Singular Learning Theory, as outlined in his book \"Algebraic Geometry and Statistical Learning Theory\". Watanabe\'s key insight to studying singular models was to use a deep result in algebraic geometry known as Hironaka\'s Resolution of Singularities. This result allows him to reparametrize the model in a normal form so that central limit theorems can be applied. In the second half of the talk, we discuss new algebraic methods where we define fiber ideals for discrete/Gaussian models which are singular. We show how to approximate marginal likelihood integrals for these models using Newton polyhedra.