The linearization theorem for proper Lie groupoids, whose prove was completed by Zung a few years ago, generalizes various results such as Ehresmann theorem for submersions, Reeb stability for foliations, and the Tube Theorem for proper actions. In a work in progress with R. Fernandes we show that this linearization can be achieved by means of the exponential flow of certain metrics, providing both a stronger theorem and a simpler proof. In this talk I will recall the classic linearization theorems, discuss its groupoid formulation, and present our work on riemannian structures for Lie groupoids.