  We formulate the notion of \emph{typical boundedness} of torsion on a family
of abelian varieties defined over number fields. This means that the torsion
subgroups of elements in the family can be made uniformly bounded by removing
from the family all abelian varieties defined over number fields of degree
lying in a set of arbitrarily small density. We show that for each fixed $g$,
torsion is typically bounded on the family of all $g$-dimensional CM abelian
varieties. We show that torsion is \emph{not} typically bounded on the family
of all elliptic curves, and we establish results -- some unconditional and some
conditional -- on typical boundedness of torsion of elliptic curves for which
the degree of the $j$-invariant is fixed.
