  We introduce nonassociative geometric objects generalising naturally Lie
groupoids, called (smooth) \emph{quasiloopoids} and \emph{loopoids}. We prove
that the tangent bundles of smooth loopoids is canonically a smooth loopoid
(which is nontrivial in case of loopoids) and this is untrue for the cotangent
bundles. After providing a few natural constructions, we show how the Lie-like
functor associates with these objects skew-algebroids and almost Lie
algebroids, respectively, and how discrete mechanics on Lie groupoids can be
reformulated in the nonassociative case.
