Abstract: A log Calabi Yau manifold U is a complex manifold admitting a normal crossing compactification such that there is a holomorphic volume form on U with simple poles at the boundary. The Strominger–Yau–Zaslow formulation of mirror symmetry as duality of Lagrangian torus fibrations applies in this context. An algebraic version of the SYZ construction due to Gross and Siebert leads to an explicit description of the mirror as the spectrum of an algebra defined using tropical geometry, which we conjecture may be defined using Gromov–Witten invariants and coincides with the degree 0 symplectic cohomology of U. This algebra comes with a canonical vector space basis which in the special case that U is holomorphic symplectic solves conjectures of Fomin–Zelevinsky and Fock–Goncharov on cluster algebras. Joint work with Gross, Keel, Kontsevich, and Siebert.