Significant amounts of data need to be transferred in order to optimize the operation of power grids. The development of advanced metering and control infrastructure ensures a growth in the amount of data transferred within smart grids. Data compression is a strategy to reduce the burden. This paper presents current challenges in the field of time-series data compression. This paper’s novel contribution is the division of data in smart grids to real-time data used for control purposes and big data sets used for non-time-critical analysis of the system. Both of these two applications have different requirements for effective compression. Currently used algorithms are listed and described with their advantages and drawbacks for both of these applications. Details needed for the implementation of an algorithm were also provided. Comprehensive analysis and comparison are intended to facilitate the design of a data compression method tailored for a particular application. An important contribution is the description of the influence of data compression methods on cybersecurity, which is one of the major concerns in modern power grids. Future work includes the development of adaptive compression methods based on artificial intelligence, especially machine learning and quantum computing. This review will offer a solid foundation for the research and design of data compression methods.