  The renormalization group (RG) method is one of the singular perturbation
methods which is used in search for asymptotic behavior of solutions of
differential equations. In this article, time-independent vector fields and
time (almost) periodic vector fields are considered. Theorems on error
estimates for approximate solutions, existence of approximate invariant
manifolds and their stability, inheritance of symmetries from those for the
original equation to those for the RG equation, are proved. Further it is
proved that the RG method unifies traditional singular perturbation methods,
such as the averaging method, the multiple time scale method, the (hyper-)
normal forms theory, the center manifold reduction, the geometric singular
perturbation method and the phase reduction. A necessary and sufficient
condition for the convergence of the infinite order RG equation is also
investigated.
