Asymptotic safety of Yukawa systems

Different Yukawa systems in three and four dimensions are investigated. The four-dimensional systems are toy models and are plagued with the still unresolved triviality and hierarchy problem of the Standard Model Higgs sector. We use the functional renormalisation group equations and construct asymptotic safety scenarios for the four-dimensional models. This was recently done in a simple Yukawa system. In this thesis we expand this model and include a left-right asymmetry. For the three-dimensional model we investigate the critical behaviour of a second-order phase transition to a chiral-symmetry broken phase. The critical behaviour is investigated in terms of critical exponents.