  We prove that that the homotopy type of the path connected component of the
identity in the contactomorphism group is characterized by the homotopy type of
the diffeomorphism group plus some data provided by the topology of the formal
contactomorphism space. As a consequence, we show that every connected
component of the space of Legendrian long knots in $\R^3$ has the homotopy type
of the corresponding smooth long knot space. This implies that any connected
component of the space of Legendrian embeddings in $\NS^3$ is homotopy
equivalent to the space $K(G,1)\times\U(2)$, with $G$ computed by A. Hatcher
and R. Budney. Similar statements are proven for Legendrian embeddings in
$\R^3$ and for transverse embeddings in $\NS^3$. Finally, we compute the
homotopy type of the contactomorphisms of several tight $3$-folds: $\NS^1
\times \NS^2$, Legendrian fibrations over compact orientable surfaces and
finite quotients of the standard $3$-sphere. In fact, the computations show
that the method works whenever we have knowledge of the topology of the
diffeomorphism group. We prove several statements on the way that have interest
by themselves: the computation of the homotopy groups of the space of
non-parametrized Legendrians, a multiparametric convex surface theory, a
characterization of formal Legendrian simplicity in terms of the space of tight
contact structures on the complement of a Legendrian, the existence of common
trivializations for multi-parametric families of tight $\R^3$, etc.
