  We consider the following combinatorial question. Let $$ S_0 \subset S_1
\subset S_2 \subset ...\subset S_m $$ be nested sets, where #$(S_i) = i$. A
move consists of altering one of the sets $S_i$, $1 \le i \le m-1$, in a manner
so that the nested condition still holds and #$(S_i)$ is still $i$. Our goal is
to find a sequence of moves that exhausts through all subsets of $S_m$ (other
than the initial sets $S_i$) with no repeats. We call this "Gray-coding through
nested sets" because of the analogy with Frank Gray's theory of exhausting
through integers while altering only one bit at a time. Our main result is an
efficient algorithm that solves this problem. As a byproduct, we produce new
families of cyclic Gray codes through binary $m$-bit integers.
