  We introduce nonassociative geometric objects generalising naturally Lie
groupoids and called (smooth) quasiloopoids and loopoids. We prove that the
tangent bundles of smooth loopoids are canonically smooth loopoids again (it is
nontrivial in the case of loopoids). We show also that this is not true if the
cotangent bundles are concerned. After providing a few natural constructions,
we show how the Lie-like functor associates with loopoids skew-algebroids and
almost Lie algebroids and how discrete mechanics on Lie groupoids can be
reformulated in the nonassociative case.
