  In this thesis the two-particle-irreducible (2PI) formalism is investigated
with several applications, particular emphasis on renormalizability.
  In the O(N) symmetric scalar quantum field theory formulated with auxiliary
fields it is pointed out that, statements recently appeared in the literature
which raised doubts on renormalizability, are wrong. Counterterms are
constructed in details in the NLO of the large-N expansion. The
renormalizability is demonstrated at the same level of the approximation also
with eliminating the auxiliary field.
  In the one component \phi^4 model (at T=0) a method using renormalization
conditions is developed, which gives directly finite equations for the 1- and
2-point functions. This solves the problems of numerical precision of the
counterterm formulated renormalization. In the two-loop 2PI approximation the
equivalence of the new method and the procedure formulated with counterterms is
shown, both analytically and numerically. For the counterterm method also a new
algorithm is developed, which accelerates the numerical convergence.
  In the U(N)xU(N) model, an approximate solution of the field equations in the
large-N limit is found in the broken phase, for a condensate proportional to
the unit matrix. The approximation assumes the presence of a mass-hierarchy
between the scalar and pseudoscalar fields. The validity of the latter
assumption is investigated in the whole parameter space numerically. The
stability of the solution against a more general condensate is checked. It is
shown that, there exists a region of the parameter space, where the condensate
corresponding to the "8" direction of the SU(3) classification can arise
spontaneously. This solution is found to be metastable, but could transform
into the true ground state, when a moderate external source is applied along
the "8" direction.
