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.

15 pages + 4 tables; Submitted to ApJ