  Metrics on Lie groupoids and differentiable stacks have been introduced
recently, extending the Riemannian geometry of manifolds and orbifolds to more
general singular spaces. Here we continue that theory, studying stacky curves
on Riemannian stacks, measuring their length using stacky metrics, and
introducing stacky geodesics. Our main results show that the length of stacky
curves measure distances on the orbit space, characterize stacky geodesics as
locally minimizing curves, and establish a stacky version of Hopf-Rinow
Theorem. We include a concise overview that bypasses nonessential
technicalities, and we lay stress on the examples of orbit spaces of isometric
actions and leaf spaces of Riemannian foliations.
