As a result of quickly growing data, a digital transmission system is required to deal with the challenge of acquiring signals at a very high sampling rate, Fortunately, the CS (Compressed Sensing or Compressive Sensing) theory, a new concept based on theoretical results of signal reconstruction, can be employed to exploit the sparsity of the received signals. Then, they can be adequately reconstructed from a set of their random projections, leading to dramatic reduction in the sampling rate and in the use of ADC (Analog-to-Digital Converter) resources. The goal of this article is provide an overview of the basic CS theory and to survey some important compressed sensing applications in wireless communications.