Millimeter-wave (mmWave) communication and network densification hold great promise for achieving high-rate communication in next-generation wireless networks. Cloud radio access network (CRAN), in which low-complexity remote radio heads (RRHs) coordinated by a central unit (CU) are deployed to serve users in a distributed manner, is a cost-effective solution to achieve network densification. However, when operating over a large bandwidth in the mmWave frequencies, the digital fronthaul links in a CRAN would be easily saturated by the large amount of sampled and quantized signals to be transferred between RRHs and the CU. To tackle this challenge, we propose in this paper a new architecture for mmWave-based CRAN with advanced lens antenna arrays at the RRHs. Due to the energy focusing property, lens antenna arrays are effective in exploiting the angular sparsity of mmWave channels, and thus help in substantially reducing the fronthaul rate and simplifying the signal processing at the multi-antenna RRHs and the CU, even when the channels are frequency-selective. We consider the uplink transmission in a mmWave CRAN with lens antenna arrays and propose a low-complexity quantization bit allocation scheme for multiple antennas at each RRH to meet the given fronthaul rate constraint. Further, we propose a channel estimation technique that exploits the energy focusing property of the lens array and can be implemented at the CU with low complexity. Finally, we compare the proposed mmWave CRAN using lens antenna arrays with a conventional CRAN using uniform planar arrays at the RRHs, and show that the proposed design achieves significant throughput gains, yet with much lower complexity.