I'm far from expect in this topic, but I've overheard a lot of number theory talk over the yearshere's my attempt.
As algebraic geometry matured, people learned that one of the deep reasons for importance of Galois group is that it's the fundamental group
pi_1(Spec F) (depending on the definition, sometimes this is true only after a profinite completio)completion, I'll omit this fine point further below). This allows us to use the whole apparatus of topology as well as the geometric intuition.
Moreover, if we restrict ourselves to the *Abelian part of Galois group, its structure has been completely established by the class field theory. In other words, all one-dimensional representations of
Gal Q are known. That begs a natural question about higher-dimensional reps — indeed this goes on by the name of Langlands program.
One more topic in the discussion of the structure of
Gal F is about the specific (conjugacy classes) of operators that live there, called Frobenius operators. The amazing thing about them is that they behave, formally, in a way similar to the knot operators in the fundamental group of a threefold without knots. This image is reinforced by the computations of the dimension of
Spec Z that give an answer of 3, providing fruitful connections to the theory of real 3-manifolds, knots and their L-functions.
Finally, here's the direction that was explored by many people and was probably made famous through Grothendieck's work. Consider a variety
Z := P^1-(0, 1, \infty), the projective plane without three points. It is really interesting as it makes sense over any field. Now suppose we consider it as a scheme over Q and ask for its fundamental group. Then one would have an exact sequence, if I'm not mistaken,
which implies, by standard reasoning, that
Gal Q acts on
pi_1(Z/\bar Q), which is a very straightforward group — it's simply a normal topological fundamental group of a plane without two points, so it's freely generated by two loops. Grothendieck called it a cartographic group and related it to dessin's d'enfants. The standard references are very well-written in T.'s answers and there are hopefully more questions on this topic coming to MathOverflow ;)
As you see, this is a very interesting direction that we should continue to explore. It's One more thing I like about it is that it's also very suitable to MathOverflow format, since there are both many specific questions as well as opportunities to write reviews and to produce new results by collaboration.