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This paper presents investigations of free vibration of anisotropic plates of different geometrical shapes and generally restrained boundaries. The existence and uniqueness of weak solutions of boundary value problems and eigenvalue problems which correspond to the statical and dynamical behaviour of the mentioned plates is demonstrated. It is determined that when the plates have corner points formed by the intersection of edges free or elastically restrained against translation,the corresponding bilinear forms maintain the V-ellipticity property. Also, an analytical formulation, based on the Ritz method and polynomial expressions as approximate functions for analysing the free vibrations of laminated plates with smooth and non-smooth boundary with non-classical edge supports is presented. Numerical results are presented for circular, elliptical and trapezoidal plates for different boundary conditions and material properties.