The author first proves a volume comparison theorem for small balls in calibrated submanifolds of Riemannian manifolds whose sectional curvature is bounded from above. This yields an apriori bound on the diameter of calibrated submanifolds in a prescribed homology class. A Kähler manifold \(M^{2n}\) with a nowhere vanishing holomorphic \((n,0)\)-form is called almost Calabi-Yau. A special Lagrangian fibration for \(M^{2n}\) is a surjective map \(\alpha: M\to S\) onto a topological space \(S\) containing an open subset \(S_0\subset S\) such that the following two conditions are satisfied.
1. \(S_0\) is a smooth \(n\)-dimensional manifold and \(M_0=\alpha^{-1}(S_0) \to S_0\) a smooth fibration whose fibres are special Lagrangian tori in \(M\).
2. \(M\setminus M_0\) is contained in the image of a smooth map from a compact \((2n-2)\)-dimensional manifold \(N\) into \(M\). An example of a special Lagrangian fibration on an almost Calabi-Yau manifold (Borcea-Voisin threefold) is constructed in the paper.