  Analytic expressions for the statistics of peaks of random fields with weak
non-Gaussianity are provided. Specifically, the abundance and spatial
correlation of peaks are represented by formulas which can be evaluated only by
virtually one-dimensional integrals. We assume the non-Gaussianity is weak
enough such that it is represented by linear terms of the bispectrum. The
formulas are formally given in $N$-dimensional space, and explicitly given in
the case of $N=1,2,3$. Some examples of peak statistics in cosmological fields
are calculated for the cosmic density field and weak lensing field, assuming
the weak non-Gaussianity is induced by gravity. The formulas of this paper
would find a fit in many applications to statistical analyses of cosmological
fields.
