Abstract A dynamic geometrically non-linear third-order theory of multilayered anisotropic and sandwich shallow shells featuring damaged interfaces is presented. The theory (i) accounts for an arbitrary distribution of the tangential displacements through the shell thickness, (ii) fulfils a priori the continuity conditions on the transverse shear stresses at the interfaces between the layers and their vanishing on the top and bottom surfaces, (iii) allows for jumps in the in-plane displacements when interlayer slips are present, (iv) incorporates the initial geometric imperfections. The pertinent equations of motion and variationally consistent boundary conditions are derived by means of the virtual work principle. Kirchhoff-Love's and first-order shallow shell models are obtained, among others, as special cases. A numerical assessment of the relative merits of the developed shallow shell model, of the effects of interface damage, shallowness parameter and degree of movability of the edge on the non-linear response under transverse loading and a uniform temperature change of sandwich and five-layered, symmetric cross-ply cylindrical panels is presented. The developed theory can contribute to a better understanding of the load-carrying capacity and failure of shell structures exhibiting damaged interfaces.