  We develop the theory of multiplicative Ehresmann connections for Lie
groupoid submersions covering the identity, as well as their infinitesimal
counterparts. We construct obstructions to the existence of such connections,
and we prove existence for several interesting classes of Lie groupoids and Lie
algebroids, including all proper Lie groupoids. We show that many notions from
the theory of principal bundle connections have analogues in this general
setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc.
In [17] we provide a non-trivial application of the results obtained here to
construct local models in Poisson geometry and to obtain linearization results
around Poisson submanifolds.
