  In this paper we investigate the problem of the distributivity of Kurepa
trees. We show that it is consistent that there are Kurepa trees and for every
Kurepa tree there is a small forcing notion which adds a branch to it without
collapsing cardinals. On the other hand, we derive a proper forcing notion for
making an arbitrary Kurepa tree into a non-distributive tree without collapsing
$\aleph_1$ and $\aleph_2$.
