These are notes from talks of the authors on some explicit examples of
families of Calabi-Yau threefolds which are parametrised by a Shimura variety.
We briefly review the periods of Calabi-Yau threefolds and we discuss a recent
result on Picard-Fuchs equations for threefolds which are hypersurfaces with
many automorphisms. Next various examples of families parametrised by Shimura
varieties are given. Most of these are due to J.C. Rohde. The examples with an
automorphism of order three are given in some detail. We recall that such
families do not have maximally unipotent monodromy and that the Shimura
varieties in these cases are ball quotients.Comment: 15 page
