  We assume there to be precisely three left-handed neutrino states whose
Majorana masses are generated by an unspecified mechanism. Were CP conserved,
the symmetric neutrino mass matrix M would be real and all six of its distinct
entries could be experimentally determined. But CP is not conserved so that M
is likely to be complex. As a result, not all nine of its
convention-independent real parameters can be determined without an appeal to
theory. Thus we examine the possibility that a restricted class of neutrino
mass matrices may suffice to describe current data, namely those complex
symmetric matrices several of whose entries vanish. We find that there are
seven acceptable textures with two independent zeroes, and we explore their
contrasting phenomenological implications. Textures with more than two
independent zeroes appear to be excluded by experiment.
