This survey paper presents a collection of the most important algorithms for the well-known Traveling Salesman Problem (TSP) using Self-Organizing Maps (SOM). Each one of the presented models is characterized by its own features and advantages. The modes are compared to each other to find their differences and similarities. The models are classified in two basic categories, namely the enriched and hybrid models. For each model we present information regarding its performance, the required number of iterations, as well as the number of neurons that are capable of solving the TSP problem. Based on the experimental results, the best model is identified for different occasions. The paper is a good starting point for anyone who is interested in solving TSP with SOM and desires to grasp a lot about this renowned problem.