  We describe a new formulation of the functional renormalization group (RG)
for interacting fermions within a Wilsonian momentum-shell approach. We show
that the Luttinger-Ward functional is a fixed point of the RG, and derive the
infinite hierarchy of flow equations satisfied by the two-particle-irreducible
(2PI) vertices. In the one-loop approximation, this hierarchy reduces to two
equations that determine the self-energy and the 2PI two-particle vertex
$\Phi^{(2)}$. Susceptibilities are calculated from the Bethe-Salpeter equation
that relates them to $\Phi^{(2)}$. While the one-loop approximation breaks down
at low energy in one-dimensional systems (for reasons that we discuss), it
reproduces the exact results both in the normal and ordered phases in
single-channel (i.e. mean-field) theories, as shown on the example of BCS
theory. The possibility to continue the RG flow into broken-symmetry phases is
an essential feature of the 2PI RG scheme and is due to the fact that the 2PI
two-particle vertex, contrary to its 1PI counterpart, is not singular at a
phase transition. Moreover, the normal phase RG equations can be directly used
to derive the Ginzburg-Landau expansion of the thermodynamic potential near a
phase transition. We discuss the implementation of the 2PI RG scheme to
interacting fermion systems beyond the examples (one-dimensional systems and
BCS superconductors) considered in this paper.
