  We generalize Calabi-Yau 3-folds from the special Lagrangian perspective.
More precisely, we study SU(3)-structures which admit as "nice" a local special
Lagrangian geometry as the flat $\mathbf{C}^3$ or a Calabi-Yau structure does.
The underlying almost complex structure may not be integrable. Such
SU(3)-structures are called {\it admissible}. Among these, we are particularly
interested in a subclass of SU(3)-structures called {\it nearly Calabi-Yau}. We
discuss the local generalities of admissible SU(3)-structures and nearly
Calabi-Yau structures in Cartan's sense. Examples of nearly Calabi-Yau but
non-Calabi-Yau structures as well as other admissible SU(3)-structures are
constructed from the twistor spaces of self-dual Einstein 4-manifolds. Two
classes of examples of complete special Lagrangian submanifolds are found by
considering the anti-complex involution of the underlying SU(3)-structures. We
finally show that the moduli space of compact special Lagrangian submanifolds
in a nearly Calabi-Yau manifold behaves in the same way as the moduli in the
Calabi-Yau case.
