We consider variants of the classic bin packing and multiple knapsack
problems, in which sets of items of {\em different classes (colors)} 
need to be placed in bins; the items may have different sizes and values.
Each bin has a limited capacity, and a bound on the number of distinct
classes of items it can accommodate. In the {\em class-constrained multiple
knapsack (CCMK)} problem, our goal is to maximize the total value of packed
items, whereas in the {\em class-constrained bin-packing (CCBP)}, we seek to 
minimize the number of (identical) bins, needed for packing all the items. 

We give a {\em polynomial time approximation scheme} (PTAS) for CCMK and 
a dual PTAS for CCBP. We also show that the 0-1 class-constrained knapsack  
admits a {\em fully} polynomial time approximation scheme, even when the
number of distinct colors of items depends on the input size.
Finally, we introduce the {\em generalized class-constrained packing
problem} ({\em GCCP}), where each item may have more than one color. We
show that GCCP is APX-hard, 
already for the case of a single knapsack, where all items have the 
same size and the same value.

Our optimization problems have several important applications, including
storage management for multimedia systems, production planning,
and multiprocessor scheduling.