In this thesis ultracold quantum gases are studied with field theoretical methods. The quantum mechanical three-body problem is solved exactly and the recently experimentally observed universality of the three-body parameter is explained. By means of the functional renormalization group (fRG) the four-body problem is analyzed and it is shown that no additional four-body parameter has to be introduced. The second part of this thesis studies a single impurity immersed in a majority Fermi sea. We develop a new, numerical fRG technique which allows the computation of Green’s functions with full frequency and momentum dependence. Our results are in excellent agreement with recent experiments and exemplify the intricate interplay between few- and many-body physics in ultracold quantum gases.