Abstract. Let H be a noncommutative regular projective curve over a perfect field k. We study global and local properties of the Auslander-Reiten translation τ and give an explicit description of the complete local rings, with the involvement of τ . We introduce the τ -multiplicity eτ (x), the order of τ as a functor restricted to the tube concentrated in x. We obtain a local-global principle for the (global) skewness s(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 H which fix all objects, is determined by the points x with eτ (x) > 1. Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(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 H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the τ -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.