This paper introduces the synchrosqueezed curvelet transform as an optimal tool for two-dimensional mode decomposition of wavefronts or banded wave-like components. The syn- chrosqueezed curvelet transform consists of a generalized curvelet transform with application depen- dent geometric scaling parameters, and a synchrosqueezing technique for a sharpened phase space representation. In the case of a superposition of banded wave-like components with well-separated wave-vectors, it is proved that the synchrosqueezed curvelet transform is capable of recognizing each component and precisely estimating local wave-vectors. A discrete analogue of the continuous transform and several clustering models for decomposition are proposed in detail. Some numeri- cal examples with synthetic and real data are provided to demonstrate the above properties of the proposed transform.