Coding theory has several applications in Genetics and Bioengineering. This book concentrates on a specific application from Computational Biology. This concerns the construction of new DNA codes which satisfy certain combinatorial constraints, using an alphabet of four symbols. The interest in these codes arises because it is possible to synthesise short single strands of DNA known as oligonucleotides. The codes can be useful in the design of these oligonucleotides. For example, the codes are used in DNA computing, as bar codes in molecular libraries and in microarray technologies. Magma software is applied to the construction of DNA codes. The constraints are specified to avoid unwanted hybridizations and to ensure uniform melting temperatures. Codes constructed in this book are derived from linear codes over GF(4) and Z4 and additive codes over GF(4). In many cases codes are obtained which are larger than the best codes currently known. Generator polynomials are given for the codes constructed. Coset leaders are also given in cases where cosets of linear codes are used. Thus it is possible for the reader to construct the codes without repeating the work presented in the book.