Intermediate or fractional statistics appear as theoretical possibilities when quantizing systems of identical particles in one and two dimensions. In the report these possibilities are reviewed, and basic properties of two-dimensional anyon systems are discussed. An alternative approach based on the representation of a fundamental set of symmetric observables is examined. The approach is related to effects in physical systems, in particular for vortex dynamics and particle motion in magnetic fields. Reformulations in terms of singular ``statistics interactions`` are discussed for the two approaches and the connection between Berry phases and statistics phases is examined. 43 refs.