  This is the author's PhD thesis. Two main sections address various aspects of
mirror symmetry for compact Calabi-Yau threefolds and the roles that
classically modular varieties play in string theory compactifications. The main
results include a study, and finding an application to the higher genus
problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa
invariants; provision of a new class of solutions to the supersymmetric flux
vacuum equations which have elsewhere been conjectured to give weight-two
modular manifolds; provision of two new conjectural examples of weight-four
modular varieties (rank-two attractors); and discussion of a set of numerical
relations between infinite sums of Gromov-Witten invariants and critical
L-values.
