Compressed sensing (CS) is a mathematical framework that reconstructs data from highly undersampled measurements. To gain acceleration in acquisition time, CS has been applied to MRI and has been demonstrated on diverse MRI methods. This review discusses the important requirements to qualify MRI to become an optimal application of CS, namely, sparsity, pseudo-random undersampling, and nonlinear reconstruction. By utilizing concepts of transform sparsity and compression, CS allows acquisition of only the important coefficients of the signal during the acquisition. A priori knowledge of MR images specifically related to transform sparsity is required for the application of CS. In this paper, Section I introduces the fundamentals of CS and the idea of CS as applied to MRI. The requirements for application of CS to MRI is discussed in Section II, while the various acquisition techniques, reconstruction techniques, the advantages of combining CS and parallel imaging, and sampling mask design problems are discussed in Section III. Numerous applications of CS in MRI due to its ability to improve imaging speed are reviewed in section IV. Clinical evaluations of some of the CS applications recently published are discussed in Section V. Section VI provides information on available open source software that could be used for CS implementations.