Shimura curves over Q parameterize abelian surfaces with quaternion actions, but explicit projective models for them are hard to find, partly because they exist only when level structure is added. However the associated universal families exist with much less level structure. We give in several examples a description of these families as very special families of elliptic surfaces. For these we need the Nikulin's theory of discriminant forms (and the Torelli Theorem for K3's). This is joint work with A. Besser.