  We construct a new family of exact quantum field theories modeled on
hyperbolic geometry, called {\it quantum hyperbolic field theories} (QHFTs).
The QHFTs are defined for a $(2+1)$-bordism category based on the set of
compact oriented 3-manifolds $Y$, equipped with properly embedded framed links
$L_\Ff$ and with flat connections $\rho$ of principal $PSL(2,\C)$-bundles over
$Y \setminus L_\Ff$, with arbitrary holonomy at the link meridians. A main
point is the introduction of new parameters for the space of all
$PSL(2,\C)$-characters of a punctured surface. Each QHFT associates to a triple
$(Y,L_\Ff,\rho)$ as above with parametrized boundary components a tensor, which
is generically holomorphic w.r.t. the parameters for the restriction of $\rho$
to $\partial Y \setminus L_\Ff$. This gives new numerical invariants of
3-manifolds, such as Chern-Simons invariants of $PSL(2,\mc)$-characters of
arbitrary link complements, or quantum invariants of compact hyperbolic cone
manifolds. Also, for any $PSL(2,\mc)$-character of a surface of finite
topological type, we obtain new conjugacy classes of linear representations of
the mapping class group. Finally, we discuss some evidences showing that the
QHFTs are pertinent to 3D gravity.
