  The aim of this article is to report on recent progress in understanding
mirror symmetry for some non-complete intersection Calabi-Yau threefolds. We
first construct four new smooth non-complete intersection Calabi-Yau threefolds
with h^{1,1}=1, whose existence was previously conjectured by C. van Enckevort
and D. van Straten. We then compute the period integrals of candidate mirror
families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new
Calabi-Yau threefolds. The Picard-Fuchs equations coincide with the expected
Calabi-Yau equations. Some of the mirror families turn out to have two
maximally unipotent monodromy points.
