This work explores and extends the concept of applying compressed sensing to MRI. Asuccessful CS reconstruction requires incoherent measurements, signal sparsity, and a nonlinearsparsity promoting reconstruction. To optimize the performance of CS, the acquisition, thesparsifying transform and the reconstruction have to be adapted to the application of interest.This work presents new approaches for sampling, signal sparsity and reconstruction, which areapplied to three important applications: dynamic MR imaging, MR parameter mapping andchemical-shift based water-fat separation.The methods presented in this work allow to more fully exploit the potential of compressedsensing to improve imaging speed. Future development of these methods, and combination withexisting techniques for fast imaging, holds the potential to improve the diagnostic quality ofexisting clinical MR imaging techniques and to open up opportunities for entirely new clinicalapplications of MRI.