  In this thesis we develop a Chabauty-Kim theory for the relative completion
of motivic fundamental groups, including Selmer stacks and moduli spaces of
admissible torsors for the relative completion of the de Rham fundamental
group. On one hand, this work generalizes results of Kim (and therefore
Chabauty) in the unipotent case by adding a reductive quotient of the
fundamental group. From this perspective, the addition of a reductive part
allows one to apply Chabauty-type methods to fundamental groups with trivial
unipotent completion, such as $SL_2(\mathbb{Z})$.
  On the other hand, the unipotent part provides a natural extension of the
recent work of Lawrence and Venkatesh. We show that their concern with the
centralizer of Frobenius goes away as one moves up the unipotent tower and away
from the reductive world of flag varieties and the Gauss-Manin connection. One
is tempted to hope that the relative completion will provide a unified proof of
Mordell's conjecture that takes advantage of the two methods. Toward this end,
we apply our work to the Legendre family on the projective line minus three
points, a particular example where the method of Lawrence and Venkatesh fails.
