We study approximation of some well-known network design problems such as traveling salesman problem (for both minimization and maximization versions) and min steiner tree, by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time.