In this talk, we will discuss the dichotomy recurrence/transience and ballisticity in the context of a class of random walks that build their on domain. At each step of the walker on its domain, a random number of new vertices is attached to the walker's position. We will present distributional conditions over this random number of new vertices for which we observe distinct sharp behavior on the walker. We also discuss structural results, that is, when this process is seen as a random graph model, the walker is capable of generating tree whose degree distribution approaches a power-law.