  We introduce the notions of a differentiable groupoid and a differentiable
stratified groupoid, generalizations of Lie groupoids in which the spaces of
objects and arrows have the structures of differentiable spaces, respectively
differentiable stratified spaces, compatible with the groupoid structure. After
studying basic properties of these groupoids including Morita equivalence, we
prove a de Rham theorem for locally contractible differentiable stratified
groupoids. We then focus on the study of the inertia groupoid associated to a
proper Lie groupoid. We show that the loop and the inertia space of a proper
Lie groupoid can be endowed with a natural Whitney B stratification, which we
call the orbit Cartan type stratification. Endowed with this stratification,
the inertia groupoid of a proper Lie groupoid becomes a locally contractible
differentiable stratified groupoid.
