  We develop the general theory for the construction of Extended Topological
Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau
quantum invariants of closed 3-manifolds. In order to do so, we introduce
relative modular categories, a class of ribbon categories which are modeled on
representations of unrolled quantum groups, and which can be thought of as a
non-semisimple analogue to modular categories. Our approach exploits a
2-categorical version of the universal construction introduced by Blanchet,
Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by
symmetric monoidal 2-functors which are defined over non-rigid 2-categories of
admissible cobordisms decorated with colored ribbon graphs and cohomology
classes, and which take values in 2-categories of complete graded linear
categories. In particular, our construction extends the family of graded
2+1-TQFTs defined for the unrolled version of quantum $\mathfrak{sl}_2$ by
Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The
non-semisimplicity of the theory is witnessed by the presence of non-semisimple
graded linear categories associated with critical 1-manifolds.
