Mirror symmetry predicts the existence of pairs of manifolds such that complex geometry on one corresponds to symplectic geometry on the other and vice-versa. In the case of Calabi-Yau manifolds, the Strominger-Yau-Zaslow conjecture gives a geometric procedure for constructing the mirror to a given manifold, by considering families of special Lagrangian tori inside it. Mirror symmetry has since then been extended to other settings, and the need for a geometric construction of the mirror in those settings therefore arises. The goal of this talk will be to describe how one can try to construct the mirror of a compact Kahler manifold whose anticanonical class is effective (for example a smooth Fano variety). For this purpose, we study moduli spaces of special Lagrangian tori, and show how the mirror superpotential arises from a count of holomorphic discs. The general features of the construction, including "quantum correction" phenomena, will be illustrated by considering a specific example: the complex projective plane.