As compared to single-element antenna, antenna array can not only achieve better radiation performance, but also be able to control the radiation beam. Aiming to achieve the same radiation performance, sparse antenna arrays attempt to minimize the number of antenna elements and reduce the mutual coupling between antenna elements. For sparse antenna array, there are multiple design algorithms to optimize the antenna location, like nonlinear optimization methods including genetic algorithm (GA) and simulated annealing. Recently, a new method named compressive sensing (CS) was proposed. CS could accurately recover the original signal, even at a sampling rate lower than the Nyquist rate. This thesis aims to design sparse antenna array that can achieve better performance compared with uniform antenna array, reducing the cost and optimizing antenna radiation patterns. Moreover, we aim to reduce the computational complexity. With CS, we start with a linear array and move on to a planar array, which is compared to the results obtained using the GA. Our results show that a sparse antenna array with 8 elements can be used to replace a uniform array with 100 elements. Next, both GA and CS can be used to optimize uniform planar antenna arrays. However, the computational time of CS is 350 times shorter than that of GA. In the future, we would like to apply CS in the design of three-dimensional (3D) sparse arrays.