  We consider the effective potential in three-dimensional models with O(N)
symmetry. For generic values of N, and in particular for the physically
interesting cases N=0,1,2,3, we determine the six-point and eight-point
renormalized coupling constants which parametrize its small-field expansion.
These estimates are obtained from the analysis of their $\epsilon$-expansion,
taking into account the exact results in one and zero dimensions, and, for the
Ising model (i.e. N=1), the accurate high-temperature estimates in two
dimensions. They are compared with the available results from other approaches.
We also obtain corresponding estimates for the two-dimensional O($N$) models.
