Mirror symmetry is a phenomenon concerning a duality between symplectic geometry and algebraic geometry, asserting that certain geometric problems on a symplectic manifold admit equivalent formulations on a related space, called a mirror space. A central class of such problems arises from counting holomorphic curves with boundary conditions, leading to invariants known as open Gromov-Witten invariants. Mirror symmetry predicts that these geometric invariants correspond to numerical invariants associated to algebraic objects, called matrix factorizations, on the mirror side. In this talk, I will discuss the basic ideas of mirror symmetry in the context of Fano manifolds. I will describe a non-Archimedean framework required for defining the numerical invariants and present results concerning their structure. This is joint work with J. Solomon.