  Let $\mathcal{H}$ be a noncommutative regular projective curve over a perfect
field $k$. We study global and local properties of the Auslander-Reiten
translation $\tau$ and give an explicit description of the complete local
rings, with the involvement of $\tau$. We introduce the $\tau$-multiplicity
$e_{\tau}(x)$, the order of $\tau$ as a functor restricted to the tube
concentrated in $x$. We obtain a local-global principle for the (global)
skewness $s(\mathcal{H})$, defined as the square root of the dimension of the
function (skew-) field over its centre. In the case of genus zero we show how
the ghost group, that is, the group of automorphisms of $\mathcal{H}$ which fix
all objects, is determined by the points $x$ with $e_{\tau}(x)>1$. Based on
work of Witt we describe the noncommutative regular (smooth) projective curves
over the real numbers; those with $s(\mathcal{H})=2$ we call Witt curves. In
particular, we study noncommutative elliptic curves, and present an elliptic
Witt curve which is a noncommutative Fourier-Mukai partner of the Klein bottle.
If $\mathcal{H}$ is weighted, our main result will be formulae for the orbifold
Euler characteristic, involving the weights and the $\tau$-multiplicities. As
an application we will classify the noncommutative $2$-orbifolds of nonnegative
Euler characteristic, that is, the real elliptic, domestic and tubular curves.
Throughout, many explicit examples are discussed.
