  We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing $\cal P$ with
congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can
be replaced with $n$ replicas of $K$, and it is completely saturated if it is
$n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent
replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced
by $n-1$ replicas of $K$ without uncovering a portion of the space, and it is
completely reduced if it is $n$-reduced for each $n\ge 1$. We prove that every
body $K$ in $d$-dimensional Euclidean or hyperbolic space admits both an
$n$-saturated packing and an $n$-reduced covering with replicas of $K$. Under
some assumptions on $K\subset \mathbb{E}^d$ (somewhat weaker than convexity),
we prove the existence of completely saturated packings and completely reduced
coverings, but in general, the problem of existence of completely saturated
packings and completely reduced coverings remains unsolved. Also, we
investigate some problems related to the the densities of $n$-saturated
packings and $n$-reduced coverings. Among other things, we prove that there
exists an upper bound for the density of a $d+2$-reduced covering of
$\mathbb{E}^d$ with congruent balls, and we produce some density bounds for the
$n$-saturated packings and $n$-reduced coverings of the plane with congruent
circles.
