TITLE: How are motives related to anabelian geometry and Galois-Teichmuller theory?
QUESTION [23 upvotes]: In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this moment, but this is what I understand:
Anabelian geometry tries to ask how much information about a variety is contained in its etale fundamental group. In particular, there exist "anabelian varieties" which should be completely determined by the etale fundamental group (up to isomorphism). The determination of these anabelian varieties is currently ongoing.
Galois-Teichmuller theory tries to understand the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ in terms of the automorphisms of the "Teichmuller tower", which is constructed as follows. We begin with the moduli stacks of curves with genus $g$ and $\nu$ marked points. These moduli stacks $\mathcal{M}_{g,\nu}$ have homomorphisms to each other, which correspond to "erasing" marked points and "gluing". The Teichmuller tower $\hat{T}_{g,\nu}$ comes from the profinite fundamental groupoids of these moduli stacks.
The theory of motives is some sort of "universal cohomology theory" in the sense that any Weil cohomology theory (which is a functor from smooth projective varieties to graded algebras over a field) factors through it. This is obtained from some process of "linearization" of algebraic varieties (considering correspondences as morphisms, followed by the process of "passing to the pseudo-abelian envelope", and formally inverting the Lefschetz motive).
Related to the theory of motives is the concept of a Tannakian category, which provides a kind of higher-dimensional analogue of Galois theory. I think the category of motives is conjectured to be a Tannakian category, via Grothendieck's standard conjectures on algebraic cycles (please correct me if I am wrong about this).
So I'm guessing Tannakian categories might provide the link between the theory of motives and anabelian geometry and Galois-Teichmuller theory (which are both related to Galois theory) that Grothendieck was talking about in Recoltes et Semailles, but I'm not really sure. Either way, the ideas are still not very clear to me, and I'd like to understand the connections more explicitly.

REPLY [24 votes]: There's a very deep connection between motives and Grothendieck-Teichmüller theory but it isn't well-understood yet.  I can't even frame it precisely in higher genus, but at least I can frame a precise conjecture in genus zero.  It has to do with motives that are connected to periods of moduli spaces on the one hand, and Grothendieck-Teichmüller being connected to automorphisms of fundamental groups of moduli spaces on the other.
So on the one hand, we have the Tannakian category $MTM$ of mixed Tate motives over Z.  Goncharov and Manin gave a construction showing how to construct motivic multiple zeta values from the cohomology of the moduli spaces $M_{0,n}$ of genus zero curves with $n$ marked points, giving rise to a Tannakian subcategory of MTM, and F. Brown subsequently proved that the motivic multiple zeta category actually fills all of $MTM$.  Take the Tannakian fundamental group of this category $MTM$, and take its (pro)-unipotent radical.  The associated graded Lie algebra is known to be freely generated by one generator in each odd weight.
Now on the other hand, take the Grothendieck-Teichmüller group, which can be identified with the outer automorphism group of the tower of all the fundamental groups of the moduli spaces $M_{0,n}$ that commute (up to inner automorphisms) with certain standard maps between these fundamental
groups coming from erasing points or subsurface inclusion (or doubling
the braid strands, if you like to think of the fundamental group of $M_{0,n}$ as a braid group). This is a priori a profinite group, but
it has a pro-unipotent version, which has an associated Lie algebra that
is isomorphic to the graded Grothendieck-Teichmüller Lie algebra $grt$ defined by three relations that are the exact additive analogues of the three defining relations in the profinite group.
Thanks to the result that "the motivic multizeta values satisfy the associator relations", together with Brown's result, we can deduce that the Tannakian fundamental Lie algebra of $MTM$ injects into the 
Grothendieck-Teichmüller Lie algebra $grt$.  The important conjecture is that these two Lie algebras are equal, even though one of them arises from a motivic construction which on the one hand "lifts" up actual real numbers (the real multizeta values, which are periods of the moduli spaces $M_{0,n}$ to mixed Tate motives, which then turn out to generate the full category of mixed Tate motives, and the other arises from the automorphism group of the fundamental groups of the moduli spaces $M_{0,n}$. 
This answer is not disjoint from Will Sawin's answer above, but expressed differently.