Among Poincare's numerous mathematical contributions is one related to  
the local theory of differential equations. Near an equilibrium point  
of a system of ordinary differential equations, one can transform the  
system to a "normal form", provided that certain "non-resonant"  
conditions are satisfied. In this talk, I will first introduce the  
concept of "resonance", and then present Poincare's theorem on his  
normal form. Finally I will talk about the extension of Poincare's  
normal form to partial differential equations, which is currently an  
active field of research.