The past three decades has witnessed substantial progress towards the application of low-rate speech coders to civilian and military communications as well as computer-related voice applications. Central to this progress has been the development of new speech coders capable of producing high-quality speech at low data rates. Most of these coders incorporate mechanisms to: represent the spectral properties of speech, provide for speech waveform matching, and "optimize" the coder's performance for the human ear. A number of these coders have already been adopted in national and international cellular telephony standards. The objective of this paper is to provide a tutorial overview of speech coding methodologies with emphasis on those algorithms that are part of the recent low-rate standards for voice applications. Although the emphasis is on the new low-rate coders, we attempt to provide a comprehensive survey by covering some of the traditional methodologies as well. The paper starts with a historical perspective and continues with a brief discussion on the speech properties and performance measures. Then I proceed with descriptions of waveform coders, linear predictive vocoders, and analysis-by-synthesis linear predictive coders. At the end the system for computer-based stenographing is presented. Quality research and ways how to improve this system will be provided.