Calibrated submanifolds are distinguished classes of minimal submanifolds and their moduli spaces are expected to play an important role in geometry, low dimensional topology and theoretical physics. Examples of these submanifolds are special Lagrangian 3-folds for Calabi-Yau, associative 3-folds and coassociative 4-folds for G_2, and Cayley 4-folds for Spin(7) manifolds. In this talk we give an introduction to calibrated geometries and a survey of recent research about the deformation theory of calibrated submanifolds inside Ricci-flat manifolds.