  The work in this article is concerned with two different types of families of
finite sets: separating families and splitting families (they are also called
"systems"). These families have applications in combinatorial search, coding
theory, cryptography, and related fields. We define and study generalizations
of these two notions, which we have named $n$-separating families and
$n$-splitting families. For each of these new notions, we outline their basic
properties and connections with the well-studied notions. We then spend the
greatest effort obtaining lower and upper bounds on the minimal size of the
families. For $n$-separating families we obtain bounds which are asymptotically
tight within a linear factor. For $n$-splitting families this appears to be
much harder; we provide partial results and open questions.
