The history of normal forms theory goes back to more than one hundred years ago, when Poinca\{r}e approached the problem of integrating nonlinear differential equations and developed normal form theory. Since the normal form theory has played a fundamental role in the study of qualitative behavior of dynamic systems Take noticed that the classical normal forms could be further simplified Methods have been developed for computing unique normal forms. Unique normal form is the simplest normal form(SNF) in its own style. In this thesi we study the simplest normal form of Hopf-zero singularity. The Hopf-zero linear degeneracy and correspondes to the simultaneous occurrence of a Hopf and steady state bifurcation of an equilibrium point. It is three dimensional singularity. It i important because of several reaso firstly, interacting oscillatory and stationary behavior yield to interesting three dimensional dynamical behavior, and secondly there are a lot of model in the real word are Hopf-zero. After a center manifold reduction if necessary, we deal with three dimensional system, with anequilibrium point having a double degeneracy corresponding to a zero and pair of pure imaginary eigenvalues.