  This is a concise introduction to the theory of Lie groupoids, with emphasis
in their role as models for stacks. After some preliminaries, we review the
foundations on Lie groupoids, and we carefully study equivalences and proper
groupoids. Differentiable stacks are geometric objects which have manifolds and
orbifolds as special instances, and can be presented as the transverse geometry
of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the
same stack, and proper groupoids are presentations of separated stacks, which
by the linearization theorem are locally modeled by linear actions of compact
groups. We discuss all these notions in detail. Our treatment diverges from the
expositions already in the literature, looking for a complementary insight over
this rich theory that is still in development.
