This paper presents a general two-dimensional approach for solving doubly-curved laminated composite shells using different kinematic expansions along the three orthogonal directions of the curvilinear shell model. The Carrera Unified Formulation (CUF) with different thickness functions along the three orthogonal curvilinear directions is applied to completely doubly-curved shells and panels, different from spherical and cylindrical shells and plates. Furthermore, the fundamental nuclei for doubly-curved structures are presented in their explicit form for the first time by the authors. These fundamental nuclei also allow to consider doubly-curved structures with variable thickness. In addition, the theoretical model includes the Murakami’s function (also known as zig-zag effect). For some problems it is useful to have an in-plane kinematic expansion which is different from the normal one. The 2D free vibration problem is numerically solved through the Local Generalized Differential Quadrature (LGDQ) method, which is an advanced version of the well-known Generalized Differential Quadrature (GDQ) method. The main advantage of the LGDQ method compared to the GDQ method is that the former can consider a large number of grid points without losing accuracy and keeping the very good stability features of GDQ method as already demonstrated in literature by the authors.