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...