In this paper we study the complexity of the decision problems about generic properties on the dynamics of finite discrete dynamical systems (fDDS). Properties are grouped into two main classes : local and global. Local properties are at most in , while global ones are at most in PSPACE. We also investigate universality (w.r.t. simulation) and we provide a constructive example of universal fDDS for the family of additive fDDS having a unique global attractor. The question of the complexity of deciding universality for a given family of fDDS is left open.