Objective: As the MRI main magnetic field rises for improved Signalto-Noise Ratio, susceptibility-induced B0-inhomogeneity increases proportionally, aggravating related artifacts. Considering only susceptibility disparities between air and biological tissue, we explore the topological conditions for which perfect shimming could be performed in a Region of Interest (ROI) such as the human brain or part thereof. Materials and Methods: After theoretical considerations for perfect shimming, spherical harmonic (SH) shimming simulations of very high degree are performed, based on a 100-subject database of 1.7-mmresolved brain fieldmaps acquired at 3 T . In addition to the whole brain, shimmed ROIs include slabs targeting the prefrontal cortex, both or single temporal lobes, or spheres in the frontal brain above the nasal sinus. Results and Discussion: We show “perfect” shimming is possible only if the ROI can be contained in a sphere that does not enclose sources of magnetic field inhomogeneity, which are gathered at the air-tissue interface. We establish a 12 Hz inhomogeneity hard shim limit at 7 T for whole brain shimming, that can only be attained at shimming degree higher than 90. On the other hand, under limited power and SH degree resources, 3D regionspecific shimming is shown to greatly improve homogeneity in critical zones such as the prefrontal cortex and around ear canals.