The maximum-likelihood decoding of convolutional codes has generally been considered impractical for other than relatively short constraint length codes, because of the exponential growth in complexity with increasing constraint length. The soft-decision minimum-distance decoding algorithm proposed in the paper approaches the performance of a maximum-likelihood decoder, and uses a sequential decoding approach to avoid an exponential growth in complexity. The algorithm also utilises the distance and structural properties of convolutional codes to considerably reduce the amount of searching needed to find the minimum soft-decision distance paths when a back-up search is required. This is done in two main ways. First, a small set of paths called permissible paths are utilised to search the whole of the subtree for the better path, instead of using all the paths within a given subtree. Secondly, the decoder identifies which subset of permissible paths should be utilised in a given search and which may be ignored. In this way many unnecessary path searches are completely eliminated. Because the decoding effort required by the algorithm is low, and the decoding processes are simple, the algorithm opens the possibility of building high-speed long constraint length convolutional decoders whose performance approaches that of the optimum maximum-likelihood decoder. The paper describes the algorithm and its theoretical basis, and gives examples of its operation. Also, results obtained from practical implementations of the algorithm using a high-speed microcomputer are presented.