Tropical geometry is a useful tool to study the Gromov-Witten type invariants, which count the number of holomorphic curves with incidence conditions. On the other hand, holomorphic discs with boundaries on the Lagrangian fibration of a Calabi-Yau manifold play an important role in the quantum correction of the mirror complex structure. In this talk, I will introduce a version of open Gromov-Witten invariants counting such discs and the corresponding tropical geometry on (log) Calabi-Yau surfaces. Using Lagrangian Floer theory, we will establish the equivalence between the open Gromov-Witten invariants with weighted count of tropical discs. In particular, the correspondence theorem implies the folklore conjecture that certain open Gromov-Witten invariants coincide with the log Gromov-Witten invariants with maximal tangency for the projective plane.