  A packing of partial difference sets is a collection of disjoint partial
difference sets in a finite group $G$. This configuration has received
considerable attention in design theory, finite geometry, coding theory, and
graph theory over many years, although often only implicitly. We consider
packings of certain Latin square type partial difference sets in abelian groups
having identical parameters, the size of the collection being either the
maximum possible or one smaller. We unify and extend numerous previous results
in a common framework, recognizing that a particular subgroup reveals important
structural information about the packing. Identifying this subgroup allows us
to formulate a recursive lifting construction of packings in abelian groups of
increasing exponent, as well as a product construction yielding packings in the
direct product of the starting groups. We also study packings of certain
negative Latin square type partial difference sets of maximum possible size in
abelian groups, all but one of which have identical parameters, and show how to
produce such collections using packings of Latin square type partial difference
sets.
