What is the asymptotic runtime of the best known TSP solving algorithm?
The worst case running time of Concorde or any other known method is exponential in the size of the input. However, sometimes heuristics or other pruning techniques are effective and you are able to solve some, even large, instances pretty quickly. You should define exactly what you mean by TSP as there are many variants, and many algorithms with different worst case run times.

See the accepted answer in this question on CSTheory to see a list of algorithms.