Quantum statistical physics is based on the notions of fermions and bosons. For example, the thermal state of a large quantum system can often be characterized as a thermal population of fermionic or bosonic quasiparticles, e.g., magnons being a textbook example for the bosonic case. Strikingly, certain interacting two-dimensional quantum models have quasiparticle excitations that are neither bosons nor fermions, but somewhere in between; they are called anyons, and they might be useful for noise-resistant topological quantum computing. The goal of the PhD work is to understand these models, to understand the physical consequences of anyonic quasiparticles, with the aim of proposing specific experiments to detect those measurable consequences.