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.