In this paper, we study contact structures on any open
$3$-manifold $V$ that is the interior of a compact $3$-manifold. To do
this, we introduce new proper contact isotopy invariants called the slope at
infinity and the division number at infinity. We first prove several
classification theorems for $T^2 \times [0, \infty)$, $T^2 \times \R$, and
$S^1 \times \R^2$ using these concepts. The only other classification
result on an open $3$-manifold is Eliashberg's classification on $\R^3.$
Our investigation uncovers a new phenomenon in contact geometry: There are
infinitely many tight contact structures on $T^2 \times [0,1)$ that cannot
be extended to a tight contact structure on $T^2 \times [0, \infty)$.
Similar results hold for $T^2 \times \R$ and $S^1 \times \R^2$. Finally, we
show that if every $S^2 \subset V$ bounds a ball or an $S^2$ end, then there
are uncountably many tight contact structures on $V$ that are not
contactomorphic, yet are isotopic. Similarly, there are uncountably many
overtwisted contact structures on $V$ that are not contactomorphic, yet are
isotopic. These uncountability results generalize work by Eliashberg when
$V = S^1 \times \R^2$.