  We introduce a natural generalization of twisted maps, called \emph{warped
maps}. While twisted maps play an important role in the study of
Deligne-Mumford stacks, warped maps are better suited for studying Artin
stacks. Heuristically, warped maps see the hidden proper-like behavior
satisfied by good moduli space maps. Specifically, we show that every arc of a
good moduli space admits a \emph{canonical} lift, in a warped sense, thereby
proving a valuative criterion for good moduli spaces. Furthermore, we prove
that warped maps to an Artin stack $\mathcal{X}$ are given by usual maps to an
auxiliary Artin stack $\mathscr{W}(\mathcal{X})$, immediately obtaining a
versatile framework for bootstrapping results about usual maps to the setting
of warped maps. As an application we obtain a motivic change of variables
formula which, given a stacky resolution of singularities $\mathcal{X} \to Y$,
canonically expresses any given motivic integral over arcs of $Y$ as a certain
motivic integral over warped arcs of $\mathcal{X}$. In particular, this yields
a McKay correspondence for linearly reductive groups.
