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The paper proposes a new approach for the recovery of missing data from phasor measurement units (PMUs). The approach is based on the application of tensor decomposition to PMU data organized as three-dimensional tensors with respect to time, location and type of variables. The organization of the PMU data is in the form of a bus-oriented data structure and a branch-oriented data structure. Two versions of the approach are introduced. The first version uses polyadic tensor decomposition and the second version uses Tucker tensor decomposition. Several case studies are conducted validating the proposed approaches including measured PMU data in the New York transmission system. Comparison with existing approaches is performed to demonstrate the improvement in missing data recovery accuracy achieved by the new approach. [ABSTRACT FROM AUTHOR]