Particles with arbitrary real spin (neither integer nor half-integer) whose statistics are intermediate between Bose and Fermi statistics are called ``anyons''. Anyons can occur in gauge theories, and the authors expect that they can only occur in gauge theories, or in theories with a hidden local gauge invariance. Anyons are not localizable in a bounded region, but may be localizable in a spacelike cone with an arbitrarily small opening angle. As an example of a relativistic quantum field theory, the authors study the Euclidean version (time is purely imaginary) of the 3-dimensional (noncompact) abelian Higgs model with a Chern-Simons term in the action. In the Higgs phase (with a broken symmetry) this model admits topological solitons, the quantum vortices, having the properties of anyons. These vortices are described by a (nonlocal) gauge-invariant interpolating field and, carrying a magnetic flux m, they have electric charge \(2\theta\) me and total angular momentum \(\theta m^ 2 mod {\mathbb{Z}}\), where e is the electromagnetic coupling constant and \(\theta\) (\(\not\in {\mathbb{Z}})\) is the coefficient in front of the Chern-Simons term. In particular, a one-anyone state \((m=\pm 1)\) has spin \(\theta\) mod \({\mathbb{Z}}\). The Green functions for the anyon fields are presented in terms of Euclidean functional integrals; the anyon states are obtained from the Green functions by the Osterwalder-Schrader reconstruction. The authors show how to construct a collision Haag-Ruelle type theory for asymptotic (in- and out) anyon states and discuss their (\(\theta)\)-statistics. The results may be relevant for the theory of the quantum Hall effect and for the models of high \(T_ c\) conductivity.