Abstract: We develop fifth-order A-WENO finite-difference schemes based on the path-conservative central-upwind method for nonconservative one- and two-dimensional hyperbolic systems of nonlinear PDEs. The main challenges in development of accurate and robust numerical methods for the studied systems come from the presence of nonconservative products. We apply the developed schemes to the two-layer shallow water equations. We ensure that the developed schemes are well-balanced in the sense that they are capable of exactly preserving ``lake-at-rest'' steady states. We illustrate the performance of the new fifth-order schemes on a number of one- and two-dimensional examples, where one can clearly see that the proposed fifth-order schemes clearly outperform their second-order counterparts.