Abstract: Compressive sensing/compressive sampling (CS) is a novel information theory proposed recently. CS provides a new sampling theory to reduce data acquisition, which says that sparse or compressible signals can be exactly reconstructed from highly incomplete random sets of measurements. CS broke through the restrictions of the Shannon theorem on the sampling frequency, which can use fewer sampling resources, higher sampling rate and lower hardware and software complexity to obtain the required measurements. CS has been used widely in many fields including digital cameras, medical imaging, remote sensing, seismic exploration, multimedia hybrid coding, communications and structural health monitoring. This article firstly summarizes some key issues in CS, and then discusses the development process of the optimization algorithms in CS from the sparsity constraints to lowrank constraints. Lastly, several related applications of CS in remote sensing, seismic exploration are reviewed.