The large amounts of synchrophasor data obtained by Phasor Measurement Units (PMUs) provide dynamic visibility into power systems. Extracting reliable information from the data can enhance power system situational awareness. The data quality often suffers from data losses, bad data, and cyber data attacks. Data privacy is also an increasing concern. In this paper, we discuss our recently proposed framework of data recovery, error correction, data privacy enhancement, and event identification methods by exploiting the intrinsic low-dimensional structures in the high-dimensional spatialoral blocks of PMU data. Our data-driven approaches are computationally efficient with provable analytical guarantees. The data recovery method can recover the ground-truth data even if simultaneous and consecutive data losses and errors happen across all PMU channels for some time. We can identify PMU channels that are under false data injection attacks by locating abnormal dynamics in the data. The data recovery method for the operator can extract the information accurately by collectively processing the privacy-preserving data from many PMUs. A cyber intruder with access to partial measurements cannot recover the data correctly even using the same approach. A real-time event identification method is also proposed, based on the new idea of characterizing an event by the low-dimensional subspace spanned by the dominant singular vectors of the data matrix.