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Examples of $n$-dimensional Ricci flat manifolds are Riemannian manifolds whose holonomy groups $Hol(g)$ are subgroups of $SU(n)$, for $n=2m$, and subgroups of the exceptional Lie group $G_2$, for $n=7$. We call them Calabi-Yau and $G_2$ manifolds, respectively. They are also examples of manifolds with special holonomy. Calibrated submanifolds of Calabi-Yau and $G_2$ manifolds are volume minimizing in their homology classes and their moduli spaces have many important applications in geometry, topology and physics. In this talk we give an introduction to calibrated geometries and a report of recent research on the calibrations inside the manifolds with special holonomy.