Error control coding is frequently used to minimise the errors which occur naturally in the transmission and storage of digital data. Many methods for decoding such codes already exist. The choice falls mainly into two areas: hard-decision algebraic decoding, a computationally-efficient method, and soft-decision combinatorial decoding, which although more complex offers better error-correction. The work presented in this Thesis is intended to provide practical decoding algorithms which can be implemented in real systems. Soft-decision maximum-likelihood decoding of Reed-Solomon codes can be obtained by using the Viterbi algorithm over a suitable trellis. Two-stage decoding of Reed-Solomon codes is presented. It is an algorithm by which near-optimum performance may be achieved with a complexity lower than the Viterbi algorithm. The soft-output Viterbi algorithm (SOVA) has been investigated as a means of providing soft-decision information for subsequent decoders. Considerations of how to apply SOVA to multi-level codes are given. The use of SOVA in a satellite downlink channel is discussed. The results of a computer simulation, which showed a 1.8dB improvement in coding gain for only a 20% increase in decoding complexity, are presented. SOVA was also used to improve the decoding performance when applied to an RS product code. Several different decoding methods were evaluated, including cascade decoding, and a method where the row and columns were decoded alternately. A complexity measurement was developed which allows accurate comparisons of decoding complexity for trellis-based and algebraic decoders. With this technique the decoding complexity of all the algorithms implemented are compared. Also included in the comparison are the Euclidean and Berlekamp-Massey algorithms.