  Low energy hadron phenomenology involving the (u,d,s) quarks is often
approached through effective multi-quark Lagrangians with the symmetries of
QCD. A very successful approach consists in taking the four-quark
Nambu--Jona-Lasinio Lagrangian with the chiral $U_L(3)\times U_R(3)$ symmetry
in the massless limit, combined with the $U_\A(1)$ breaking six-quark flavour
determinant interaction of 't Hooft. We review the present status and some very
recent developments related to the functional integration over the cubic term
in auxiliary mesonic variables that one introduces to bosonize the system.
Various approaches for handling this functional, which cannot be integrated
exactly, are discussed: the stationary phase approximation, the perturbative
expansion, the loop expansion, their interrelation and importance for the
evaluation of the effective action. The intricate group structure rules out the
method of Airy's integral. The problem of the instability of the vacuum is
stated and a solution given by including eight-quark interactions.
