Book excerpt: Compressive sensing is a rapidly growing research field which brings a huge amount of applications in science and engineering, and leads breakthrough in the study of signal processing, imaging, tomography, and related fields therein. In this dissertation, brief summaries of compressive sensing and important results from literatures will be reviewed. Next, its applications in radar imaging and rough surface scattering problem are discussed. Various radar sensing schemes are designed andformulated in the the framework of compressive sensing, giving the estimates of measurements and target recoverability by linear or quadratic chirps. Next, the rough surface scattering problems are studied. After a review of the mathematical setup of rough surface scattering and its numerical methods, the main result for inverse rough surface scattering and numerical simulations are demonstrated. It is shown that, with Rayleigh hypothesis and suitable designed scheme, one can reconstruct the angular spectrum for the periodic rough surface and then the surface profile, with subwavelength structures. The sparsity of the angular spectra is studied as well as the Fourier series expansion of the surface profile.