  It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a
uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of
rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We
verify the conjecture for $r=1$, and more generally for Drinfeld modules having
potential good reduction at some prime above a specified prime of $K$.
Moreover, we show that within an $\Lbar$-isomorphism class, there are only
finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero
torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give an explicit
description of the possible torsion submodules. We present three methods for
proving these cases of the conjecture, and explain why they fail to prove the
conjecture in general. Finally, an application of the Mordell conjecture for
characteristic $p$ function fields proves the uniform boundedness for the
$\pp$-primary part of the torsion for rank~2 Drinfeld $\Fq[T]$-modules over a
fixed function field.
