  We present a review of the normal form theory for weakly dispersive nonlinear
wave equations where the leading order phenomena can be described by the KdV
equation. This is an infinite dimensional extension of the well-known
Poincar\'e-Dulac normal form theory for ordinary differential equations. We
also provide a detailed analysis of the interaction problem of solitary wavesas
an important application of the normal form theory. Several explicit examples
are discussed based on the normal form theory, and the results are compared
with their numerical simulations. Those examples include the ion acoustic wave
equation, the Boussinesq equation as a model of the shallow water waves, the
regularized long wave equation and the Hirota bilinear equation having a 7th
order linear dispersion.
