Signal sampling is a bridge to analog source signal and digital signal. Over the years, the base theory of signal sampling is the famous Nyquist sampling theorem, but a large amount of data generated by the waste of storage space. Compressed Sensing proposes a new sampling theory that can sample signals well below the Nyquist sampling rate. Also, the varied reconstruction algorithms of CS can faithfully reconstruct the original signal back from fewer compressive measurements. Therefore, the theory of compressed sensing has been widely used in many fields such as analog-to-signal conversion, synthetic aperture radar imaging, and speech recognition. This paper first elaborates the basic theory model of compressed sensing and focuses on the latest developments in the three aspects of signal sparse transformation, observation matrix design, and reconstruction algorithms. Then this paper also reviews several open problems in CS theory and discusses the existing difficult problems. Secondly, the application and development of compressed sensing in the imaging field are described in detail, compressive imaging technology breaks through the traditional imaging system design concept and uses hardware to achieve non-adaptive linear projection of the target image, thereby achieving the purpose of acquiring a high-precision target image with a smaller number of detectors. And in-depth discussion and analysis of current common compressive imaging systems. Finally, this paper also highlights some of the challenges and research directions in this field.