  Generalized $t$-designs, which form a common generalization of objects such
as $t$-designs, resolvable designs and orthogonal arrays, were defined by
Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\
{\bf 309} (2009), 4835--4842]. In this paper, we define a related class of
combinatorial designs which simultaneously generalize packing designs and
packing arrays. We describe the sometimes surprising connections which these
generalized designs have with various known classes of combinatorial designs,
including Howell designs, partial Latin squares and several classes of triple
systems, and also concepts such as resolvability and block colouring of
ordinary designs and packings, and orthogonal resolutions and colourings.
Moreover, we derive bounds on the size of a generalized packing design and
construct optimal generalized packings in certain cases. In particular, we
provide methods for constructing maximum generalized packings with $t=2$ and
block size $k=3$ or 4.
