Given a vector field, its simplified expression in terms of coordinate transformations is called a normal form of the dynamical system generated by the vector field. The purpose of this paper is to give a brief exposition of recent progress in the normal form theory for vector fields. More attention is paid to the algorithm for its computation as well as applications rather than the theoretical details. After explaining a brief idea about normal forms in §1, we start with the basic definition of normal forms and give a simple example in §2, and in §3 we explain a detailed algorithm for the computation of normal forms based on a simple general principle given in §2. §4 is devoted to a sketch of analysis of a laser equation as an application of normal forms to concrete differential equations.
