This tutorial paper describes various efficient implementations (published and new unpublished) of the forward and backward modified discrete cosine transform (MDCT) in the MPEG layer III (MP3) audio coding standard developed in the time period 1990-2010, including the efficient implementation of polyphase filter banks for completeness. The efficient MDCT implementations are discussed in the context of (fast) complete analysis/synthesis MDCT filter banks in the MP3 encoder and decoder. In general, for each efficient forward/backward MDCT block transforms implementation are presented: complete formulas or sparse matrix factorizations of the algorithm, the corresponding signal flow graph for the short audio block and the total arithmetic complexity as well as the useful comments related to improving the arithmetic complexity and a possible structural simplification of the algorithm. Finally, all efficient forward/backward MDCT implementations are compared both in terms of the arithmetic complexity and structural simplicity. It is important to note that almost all presented algorithms can be also used for the 2 ''-length data blocks in others MPEG audio coding standards and proprietary audio compression algorithms. (C) 2010 Elsevier B.V. All rights reserved.