We introduce new invariants of contact structures on any open 3-manifold which is the interior of a compact 3-manifold. These invariants, called the slope at infinity and the division number at infinity, lead to extensive new classification theorems for tight contact structures on S^1 \times R^2, T^2 \times R, and T^2 \times [0, \infty). As a corollary, we find an infinite number of tight contact structures which cannot be embedded into another tight contact manifold. Finally, we use the slope at infinity to show that there are an uncountable number of tight contact structures on any irreducible, open manifold with an end of nonzero genus.