  We give a function F(d,n,p) such that if K/Q_p is a degree n field extension
and A/K is a d-dimensional abelian variety with potentially good reduction,
then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the
prime-to-p torsion and to the case of purely additive reduction. These latter
bounds are applied to classify rational torsion on CM elliptic curves over
number fields of degree at most 3, on elliptic curves over Q with integral j
(recovering a theorem of Frey), and on abelian surfaces over Q with integral
moduli. In the last case, our efforts leave us with 11 numbers which may, or
may not, arise as the order of the full torsion subgroup. The largest such
number is 72.
