In several MR applications one is interested in reconstructing only a certain Region-of-Interest (ROI) of the Field-of-View (FOV) that is of anatomical or functional importance for the study. However, according to the Nyquist criterion, one is still forced to acquire Fourier (k-space) samples at a rate determined by the FOV instead of the ROI. In this paper we derive necessary and sufficient sampling conditions for recovering the ROI, for uniform and periodic non-uniform sampling, and obtain bounds on the minimal sampling density for arbitrary sampling patterns. We also extend the analysis to parallel MR imaging. The results can be applied to functional MRI, cardiac MRI, etc to reduce the total acquisition time or increase the spatial and temporal resolution, or SNR of the reconstructions.