  Several variants of the Halpern-L\"auchli Theorem for trees of uncountable
height are investigated. For $\kappa$ weakly compact, we prove that the various
statements are all equivalent. We show that the strong tree version holds for
one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the
consistency of the Halpern-L\"auchli Theorem on $d$ many $\kappa$-trees at a
measurable cardinal $\kappa$, given the consistency of a $\kappa+d$-strong
cardinal. This follows from a more general consistency result at measurable
$\kappa$, which includes the possibility of infinitely many trees, assuming
partition relations which hold in models of AD.
