Title:Numerical models and our understanding of aspherical Planetary Nebulae

Abstract: The status of numerical hydrodynamical models for Planetary Nebulae is reviewed. Since all of the numerical work is based on the interacting winds model, we start with a description of this model and give an overview of the early analytical and numerical models. Subsequently we address the numerical models which include radiation effects, first of all the ones which neglect any effects of stellar evolution. These `constant environment' models are shown to closely match typical observed nebulae, both in images and kinematic data. This shows that the basic generalized interacting winds model gives a good description of the situation in aspherical PNe. Next we discuss models that do include the effects of stellar and fast wind evolution. This introduces several new effects, the most important of which are the formation of a surrounding attached envelope, and the modification of the expansion of the nebula, which helps in creating aspherical PNe very early on in their evolution. The ionization of the slow wind also leads to a gradual smoothing out of its aspherical character, working against aspherical PNe forming in later stages. Finally we discuss some applications of the model to nebular problems.