Recent advances in speech coding algorithms and techniques based on the use of linear prediction now permit high quality voice reproduction at remarkably low bit rates. This paper reviews some of the main ideas underlying the algorithms of major interest today. The concept of removing redundancy by linear prediction is reviewed, first in the context of predictive quantization or DPCM. Then linear predictive coding, adaptive predictive coding, and vector quantization are discussed. The concepts of excitation coding via analysis-by-synthesis linear predictive coding is explained and some important enhancements such as vector sum excitations, and adaptive postfiltering are described. Low-delay coding by backward computation of LPC parameters is explained. The concept of phonetic segmentation of speech for closed-loop coding systems is also presented. Linear prediction is the key technique that underlies almost all of the important algorithms for speech coding of interest today. Finally, we discuss some recent work on nonlinear prediction of speech and its potential for the future of speech coding.