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The generalized degrees of freedom (GDoF) region of the MIMO Gaussian interference channel (IC) is obtained for the general case of an arbitrary number of antennas at each node and where the signal-to-noise ratios (SNRs) and interference-to-noise ratios vary with arbitrary exponents to a nominal SNR. The GDoF-optimal coding scheme involves message splitting and partial interference decoding and consists of linear Gaussian superposition coding of the private and common submessages that can be seen as jointly performing signal-space and signal-level interference alignment. The admissible degree of freedom (DoF)-splits between the private and common messages are also specified. A study of the GDoF region reveals various insights through the joint dependence of optimal interference management techniques at high SNR on the SNR exponents and the numbers of antennas at the four terminals. For instance, it reveals that, unlike in the scalar IC, treating interference as noise is not always GDoF-optimal even in the very weak interference regime. Moreover, while the DoF-optimal strategy that relies just on transmit/receive zero-forcing beamforming and time sharing is not GDoF optimal (and thus has an unbounded gap to capacity), the precise characterization of the very strong interference regime-where single-user DoF performance can be achieved simultaneously for both users-depends on the relative numbers of antennas at the four terminals and thus deviates from what it is in the single-input single-output case. For asymmetric numbers of antennas at the four nodes, the shape of the symmetric GDoF curve can be a “distorted W” curve to the extent that for certain multiple-input multiple-output ICs it is a “V” curve.