  Many planetary nebulae (PNe) exhibit distinctly non-axisymmetric structure in
either (i) the shape of the nebula, or (ii) in the off-centered position of the
illuminating star. By examining a large number of well resolved images of PNe
we estimate that about 30-50 percents of all PNe exhibit distinctly
non-axisymmetric structure. In this paper, we discuss how such departures from
axisymmetry can arise from the binary nature of the progenitors of the PNe. The
scenarios include (a) relatively close binaries with eccentric orbits, and (b)
longer orbital period systems with either circular or eccentric orbits. In
order to assess the fraction of PNe whose non-axisymmetric morphologies are
expected to arise in binary systems, we have carried out a detailed population
synthesis study. The expected deviations from axisymmetry are classified for
each binary and the results tabulated. We find that about 25 percents of
elliptical and 30-50 percents of bipolar PNe are expected to acquire
non-axisymmetric structure from binary interactions.
