  Baker's conjecture states that a transcendental entire function of order less
than $1/2$ has no unbounded Fatou components. It is known that, for such
functions, there are no unbounded periodic Fatou components and so it remains
to show that they can also have no unbounded wandering domains. Here we
introduce completely new techniques to show that the conjecture holds in the
case that the transcendental entire function is real with only real zeros, and
we prove the much stronger result that such a function has no orbits of
unbounded wandering domains whenever the order is less than 1. This raises the
question as to whether such wandering domains can exist for any transcendental
entire function with order less than 1.
  Key ingredients of our proofs are new results in classical complex analysis
with wider applications. These new results concern: the winding properties of
the images of certain curves proved using extremal length arguments, growth
estimates for entire functions, and the distribution of the zeros of entire
functions of order less than 1.
