  In this article, which is dedicated to my friend and colleague Boris Zilber
on the occasion of his 75th birthday, I put forward a strategy for proving his
quasiminimality conjecture for the complex exponential field. That is, for
showing that every subset of $\mathbb{C}$ definable in the expansion of the
complex field by the complex exponential function is either countable or
cocountable. In fact the strategy applies to any expansion of the complex field
by a countable set of entire functions (in any number of variables) and is
based on a certain property-an analytic continuation property-of the o-minimal
structure obtained by expanding the ordered field of real numbers by the
restrictions to compact boxes of the real and imaginary parts of the functions
in the given set.
  In a final section I discuss briefly the (rather limited) extent of our
unconditional knowledge in the area.
