  Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New
classes of (possibly unbounded) operators $t:E\to F$ are introduced and
investigated. Instead of the density of the domain $\Def(t)$ we only assume
that $t$ is essentially defined, that is, $\Def(t)^\bot=\{0\}$. Then $t$ has a
well-defined adjoint. We call an essentially defined operator $t$ graph regular
if its graph $\Graph(t)$ is orthogonally complemented in $E\oplus F$ and
orthogonally closed if $\Graph(t)^{\bot\bot}=\Graph(t)$. A theory of these
operators is developed. Various characterizations of graph regular operators
are given. A number of examples of graph regular operators are presented
($E=C_0(X)$, a fraction algebra related to the Weyl algebra, Toeplitz algebra,
Heisenberg group). A new characterization of affiliated operators with a
$C^*$-algebra in terms of resolvents is given.
