Perturbation theory and asymptotic analysis play a fundamental role in applied mathematics and dynamical systems theory. Familiar techniques for ODEs include the method of multiple timescales, averaging, WKB, and numerous others. I will discuss the relation between the theory of normal forms and some of these classical perturbation techniques and show that the former provides an algorithmic way of obtaining rigorously justifiable approximations to solutions of certain ODEs on long timescales. I will illustrate the method by analyzing synchrony in a network of van der Pol oscillators. This is joint work with S. Peles, R.E.L. DeVille, T. Harkin and T. Kaper.