It has been conjectured that Fano manifolds correspond to certain Laurent polynomials under Mirror Symmetry. This correspondence predicts that the regularized quantum period of a Fano manifold coincides with the classical period of a Laurent polynomial mirror. This correspondence is not one-to-one, as many different Laurent polynomials can have the same classical period; it should become one-to-one after imposing the correct equivalence relation on Laurent polynomials. In this thesis we introduce what we believe to be the correct notion of equivalence: this is algebraic mutation of Laurent polynomials. We also consider combinatorial mutation, which is the transformation of lattice polytopes induced by algebraic mutation of Laurent polynomials supported on them. We establish the basic properties of algebraic and combinatorial mutations and give applications to algebraic geometry, most notably to the classification of Fano manifolds up to deformation. Our main focus is on the surface case, where the theory is particularly rich.