In this paper, we develop a moduli theory of transverse structures given by calibrations on foliated manifolds, including transverse Calabi-Yau structures. We show that the moduli space of the transverse structures is a smooth manifold of finite dimension under a cohomological assumption. We also prove a local Torelli type theorem. If the foliation is taut, we can construct a Riemannian metric on the set of transverse Riemannian structures. This metric induces a distance on the moduli space of the transverse structures given by a calibration. As an application, we show the moduli space of transverse Calabi-Yau structures is a Hausdorff and smooth manifold of finite dimension.