To alleviate the burdens on the fronthaul and reduce the transmit latency, the device-to-device (D2D) communication is presented in cloud radio access networks (C-RANs). Considering dynamic traffic arrivals and time-varying channel conditions, the resource allocation in C-RANs with D2D is formulated into a stochastic optimization problem, which is aimed at maximizing the overall throughput subject to network stability, interference, and fronthaul capacity constraints. Leveraging on the Lyapunov optimization technique, the stochastic optimization problem is transformed into a delay-aware optimization problem, which is a mixed-integer nonlinear programming problem and can be decomposed into three subproblems: mode selection, uplink beamforming design, and power control. An optimization solution that consists of a modified branch and bound method as well as a weighted minimum mean square error approach has been developed to obtain the close-to-optimal solution. Simulation results validate that the D2D can improve throughput, decrease latency, and alleviate the burdens of the constrained fronthaul in C-RANs. Furthermore, an average throughput-delay tradeoff can be achieved by the proposed solution.