Spherical logarithmic quantization (SLQ) is a vector quantization method for efficiently digitizing analog signals at a high dynamic range and with very low distortion while preserving the original waveform as closely as possible. SLQ is able to operate at a low data rate of, e.g., 2 bits per sample and at a very low signal delay of about ten samples, this corresponds to approximately 200 mus for high-quality audio signals. The technique of SLQ is universally applicable (i.e., not restricted to, e.g., audio signals) and achieves an efficient digital representation of waveforms with high longterm as well as high segmental signal-to-noise ratios. The aim of this paper is to give a detailed description of the SLQ algorithm and to present simulation results on the performance of this new quantization scheme that combines several advantages. After a review of some important basic principles concerning quantization, linear prediction and multidimensional spheres, the SLQ encoder is described. To short vectors of signal samples which are represented in sphere coordinates, logarithmic quantization is applied to the radius and uniform quantization is applied to the angles. This results in the advantage of a constant signal-to-noise ratio over a very high dynamic range at a small loss with respect to the rate-distortion theory. In order to increase the signal-to-noise ratio by exploitation of correlations within the source signal, a solution for the problem of combining this vector quantization scheme with scalar adaptive differential pulse code modulation (ADPCM), i.e., ADPCM with sample by sample backward recursion is presented. Furthermore, an indexing scheme for the quantization cells covering the surface of a multidimensional unit sphere is presented and simulation results using different source signals are given.