  The method of functional renormalization is applied to the theoretical
investigation of ultracold quantum gases. Flow equations are derived for a Bose
gas with approximately pointlike interaction, for a Fermi gas with two
(hyperfine) spin components in the Bardeen-Cooper-Schrieffer (BCS) to
Bose-Einstein condensation (BEC) crossover and for a Fermi gas with three
components. The solution of the flow equations determine the properties of
these systems both in the few-body regime and in thermal equilibrium.
  For the Bose gas this covers the quantum phase diagram, the condensate and
superfluid fraction, the critical temperature, the correlation length, the
specific heat or sound propagation. The properties are discussed both for three
and two spatial dimensions. The discussion of the Fermi gas in the BCS-BEC
crossover concentrates on the effect of particle-hole fluctuations but
addresses the complete phase diagram. For the three component fermions, the
flow equations in the few-body regime show a limit-cycle scaling and the Efimov
tower of three-body bound states. Applied to the case of Lithium they explain
recently observed three-body loss features. Extending the calculations by
continuity to nonzero density, it is found that a new trion phase separates a
BCS and a BEC phase for three component fermions close to a common resonance.
  More formal is the derivation of a new exact flow equation for scale
dependent composite operators. This equation allows for example a better
treatment of bound states.
