In geometric group theory, nilpotent groups are among the better understood classes of groups. In this thesis, we will show that nilpotent groups behave in surprising ways.

First, we prove that the growth of the number of geodesics in some virtually nilpotent groups is intermediate, these are the the first examples of this type. Next, we study the horofunction boundary of some groups, highlighting important differences between nilpotent groups of class 2 and 3, both regarding the action of a group on its boundary, or the number of Busemann points. From this point on, we take an algorithmic direction, studying the membership problems to submonoids and rational subsets. We show that these two problems are not equivalent, and solve the second problem in the Heisenberg group. In the last two chapters, we study complete growth series and Green series.