Device-cloud collaborative intelligent computing, an emergent result of the development in big data, cloud computing, and edge computing, offers significant improvements in data utilization while protecting user privacy. This approach synergizes the real-time response capabilities of intelligent computing with service robustness. The study explores the application value of this computing paradigm, highlighting technical challenges such as optimizing on-device learning efficiency, mitigating overfitting with limited samples at the device, customizing on-device models, learning false associations under distributional discrepancies, and balancing communication overhead with computational efficiency. We systematically review the progress in mainstream methods within device-cloud collaborative intelligent computing, encompassing efficient computation hardware as the application cornerstone, device-centric collaborative computing, cloud-centric collaborative computing, bidirectional device-cloud collaborative computing, and trustworthy device-cloud collaborative computing. The study also summarizes applications in vertical domains such as recommendation systems, autonomous driving, security systems, and educational models. Looking toward the future of device-cloud collaborative intelligent computing, it underscores the need for focused research on cloud resource application strategies in device model personalization, multi-objective optimization algorithms for device-cloud collaboration, and optimized collaborative strategies between devices and the cloud.