We introduce a class of Laurent polynomials, called maximally mutable Laurent
polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano
varieties. A subclass of these, called rigid, are expected to correspond to
Fano varieties with terminal locally toric singularities. We prove that there
are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror
symmetry these correspond one-to-one with the 10 deformation classes of smooth
del~Pezzo surfaces. Furthermore we give a computer-assisted classification of
rigid MMLPs in three variables with reflexive Newton polytope; under mirror
symmetry these correspond one-to-one with the 98 deformation classes of
three-dimensional Fano manifolds with very ample anticanonical bundle. We
compare our proposal to previous approaches to constructing mirrors to Fano
varieties, and explain why mirror symmetry in higher dimensions necessarily
involves varieties with terminal singularities. Every known mirror to a Fano
manifold, of any dimension, is a rigid MMLP.