Shearlets were introduced as means to sparsely encode anisotropic singularities of multivariate data while providing a unified treatment of the continuous and digital realm. In this talk, recent results on the construction of compactly supported shearlet systems will be presented, in particular, showing that these shearlet frames provide optimally sparse approximations of piecewise smooth 2D as well as 3D functions. Finally, we will discuss various applications of shearlets such as image restoration, data separation and numerical PDEs.