In this paper we use Lie group actions on noncompact Riemannian
manifolds with calibrations to construct calibrated submanifolds. In
particular, if we have an $(n-1)$-torus acting on a noncompact
Calabi-Yau $n$-fold with a trivial first cohomology, then we have a
special Lagrangian fibration on that $n$-fold. We produce several
families of examples for this construction and give some applications
to special Lagrangian geometry on compact almost Calabi-Yau
manifolds. We also use group actions on noncompact $G_2$-manifolds
to construct coassociative submanifolds, and we exhibit some new
examples of coassociative submanifolds via this setup.