The large amount of synchrophasor data obtained by Phasor Measurement Units (PMUs) provides dynamic visibility of power systems. As the data is being collected from geographically distant locations facilitated by computer networks, the data quality can be compromised by data losses, bad data, and cybernetic attacks. Data privacy is also an increasing concern. This chapter, describes a common framework of methods for data recovery, error correction, detection and correction of cybernetic attacks, and data privacy enhancement by exploiting the intrinsic low-dimensional structures in the high-dimensional spatial-temporal blocks of PMU data. The developed data-driven approaches are computationally efficient with provable analytical guarantees. For instance, 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. This approach can identify PMU channels that are under false data injection attacks by locating abnormal dynamics in the data. Random noise and quantization can be applied to the measurements before transmission to compress the data and enhance data privacy.