  We introduce the universal Euler characteristic of an orbit space definable
groupoid, a class of groupoids containing cocompact proper Lie groupoids as
well as translation groupoids for semialgebraic group actions. Generalizing
results of Gusein-Zade, Luengo, and Melle-Hern\'andez, we show that every
additive and multiplicative invariant of orbit space definable groupoids with
an additional local triviality hypothesis arises as a ring homomorphism applied
to the universal Euler characteristic. This in particular includes the
$\Gamma$-orbifold Euler characteristic introduced by the first and third
authors when $\Gamma$ is a finitely presented group. For Lie groupoids with
Riemannian structures in the sense of del Hoyo-Fernandes, and for Cartan-Lie
groupoids with Riemannian structures in the sense of Kotov-Strobl, we study the
Riemannian structures induced on the suborbifolds of the inertia space given by
images of isotropy groups. We realize the $\mathbb{Z}$-orbifold Euler
characteristic as the integral of a differential form defined on the arrow
space or the space of composable pairs of arrows of the original groupoid.
