Nilpotent Lie groups play a fundamental role in the study of geometric structures and differential equations. These groups are characterized by an algebraic property known as nilpotency. The nilpotency assumption, which also reflects a condition on the Lie algebra, imposes a certain level of commutativity for the group’s structure, allowing us to explore the interplay between algebra and geometry. It paves the way for implications in geometric analysis, geometric group theory, harmonic analysis, and control theory, as well as number theory, dynamics, and representation theory. In this chapter, we delve into the world of nilpotent Lie groups.