This article provides an overview of dynamic-range compression in digital hearing aids. Digital technology is becoming increasingly common in hearing aids, particularly because of the processing flexibility it offers and the opportunity to create more-effective devices. The focus of the paper is on the algorithms used to build digital compression systems. Of the various approaches that can be used to design a digital hearing aid, this paper considers broadband compression, multi-channel filter banks, a frequency-domain compressor using the FFT, the side-branch design that separates the filtering operation from the frequency analysis, and the frequency-warped version of the side-branch approach that modifies the analysis frequency spacing to more closely match auditory perception. Examples of the compressor frequency resolution, group delay, and compression behavior are provided for the different design approaches.