Mordell (1922) proved that the rational points of an elliptic curve $E / {\bf Q}$ form a finitely-generated abelian group. It is still not known which finitely-generated abelian groups can occur as $E({\bf Q})$. Mazur (1977) proved that the possible torsion subgroups $T$ are the cyclic groups of order $1, 2, \ldots, 10$, and $12$, and the products of cyclic groups of orders $2$ and $2k$ with $k=1,2,3,4$. For each of these fifteen $T$ it is still an open problem which ranks occur. For small $T$ the current records all come from elliptic fibrations of K3 surfaces; the most recent such record is $29$ for $|T| = 1$, found a few months ago and giving the first improvement since 2006 for curves with trivial torsion. We describe how we find elliptic K3's over $\bf Q$ whose Mordell-Weil rank is as high as possible given the torsion subgroup, and how we search for fibers of even higher rank on such a surface. This work is joint with Zev Klagsbrun (CCR-La Jolla).