In this paper, we investigate resource allocation algorithm design for multicarrier non-orthogonal multiple access (MC-NOMA) systems employing a full-duplex (FD) base station for serving multiple half-duplex (HD) downlink and uplink users simultaneously. The proposed algorithm is obtained from the solution of a non-convex optimization problem for the maximization of the weighted sum system throughput. We apply monotonic optimization to develop an optimal joint power and subcarrier allocation policy. The optimal resource allocation policy serves as a system performance benchmark due to its high computational complexity. Furthermore, a suboptimal iterative scheme based on successive convex approximation is proposed to strike a balance between computational complexity and optimality. Our simulation results reveal that the proposed suboptimal algorithm achieves a close-to-optimal performance. In addition, FD MC-NOMA systems employing the proposed resource allocation algorithms provide a substantial system throughput improvement compared with conventional HD multicarrier orthogonal multiple access (MC-OMA) systems and other baseline schemes. In addition, our results unveil that FD MC-NOMA systems enable a fairer resource allocation compared with traditional HD MC-OMA systems.