The Self-Organizing Map (SOM) algorithm was introduced by the author in 1981. Its theory and many applications form one of the major approaches to the contemporary artificial neural networks field, and new technolgies have already been based on it. The most important practical applications are in exploratory data analysis, pattern recognition, speech analysis, robotics, industrial and medical diagnostics, instrumentation, and control, and literally hundreds of other tasks. In this monograph the mathematical preliminaries, background, basic ideas, and implications are expounded in a clear, well-organized form, accessible without prior expert knowledge. Still the contents are handled with theoretical rigor.