  We outline a method to compute rational models for the Hilbert modular
surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized
abelian surfaces with real multiplication by the ring of integers in
Q(sqrt{D}), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose
structure. In particular, we compute equations for all thirty fundamental
discriminants D with 1 < D < 100, and analyze rational points and curves on
these Hilbert modular surfaces, producing examples of genus-2 curves over Q
whose Jacobians have real multiplication over Q.
