Abstract The paper examines the vibration and stability of freely supported orthotropic truncated and complete conical shells with non-homogeneous material properties under uniform lateral and hydrostatic pressures. First, the basic relations have been obtained for orthotropic truncated conical shells, Young's moduli and density which vary continuously in the thickness direction. By applying the Galerkin method, the buckling pressures and lowest cyclic frequencies of truncated and complete conical shells are obtained. To verify the accuracy of the results, comparisons are made with results available in the open literature. This study also examines in detail the effects of the variations of conical shell characteristics, the effects of the non-homogeneity and the orthotropy on the critical pressures and lowest cyclic frequencies of truncated and complete conical shells.