Low-dimensional systems, i.e. physical systems where particles live in two (or even one) space dimension, exhibit exotic quantum phenonema not occurring in the three-dimensional world. The perhaps most prominent example is the possibility of anyons, particles which are neither bosons nor fermions. Thanks to modern experimental techniques, such low-dimensional systems can be realized in the lab, often in materials which are relevant for nanotechnological applications. Thus, a theoretical understanding of these quantum phenonema is of interest both from the fundamental point of view and as basic knowledge for nanotechnology. The examples I will focus on in this talk are the quantum Hall effect (occurring in a 2D electron system at strong magnetic fields) and closely related phenonema expected to occur in rapidly rotating atomic Bose condensates.