In this paper, we discuss the compression of waveforms obtained from measurements of power system quantities and analyze the reasons why its importance is growing with the advent of smart grid systems. While generation and transmission networks already use a considerable number of automation and measurement devices, a large number of smart monitors and meters are to be deployed in the distribution network to allow broad observability and real-time monitoring. This situation creates new requirements concerning the communication interface, computational intelligence and the ability to process data or signals and also to share information. Therefore, a considerable increase in data exchange and in storage is likely to occur. In this context, one must achieve an efficient use of channel communication bandwidth and a reduced need of storage space for power system data. Here, we review the main compression techniques devised for electric signal waveforms providing an overview of the achievements obtained in the past decades. Additionally, we envision some smart grid scenarios emphasizing open research issues regarding compression of electric signal waveforms. We expect that this paper will contribute to motivate joint research efforts between electrical power system and signal processing communities in the area of signal waveform compression.51291302