The Self-Organizing Map (SOM) is widely used, easy to implement , has nice properties for data mining by providing both clustering and visual representation. It acts as an extension of the k-means algorithm that preserves as much as possible the topological structure of the data. However, since its conception, the mathematical study of the SOM remains difficult and has be done only in very special cases. In WSOM 2005, Jean-Claude Fort presented the state of the art, the main remaining difficulties and the mathematical tools that can be used to obtain theoretical results on the SOM outcomes. These tools are mainly Markov chains, the theory of Ordinary Differential Equations, the theory of stability , etc. This article presents theoretical advances made since then. In addition, it reviews some of the many SOM algorithm variants which were defined to overcome the theoretical difficulties and/or adapt the algorithm to the processing of complex data such as time series, missing values in the data, nominal data, textual data, etc.