  Using skein theory very much in the spirit of the Reshetikhin--Turaev
constructions, we define a $(3+1)$-TQFT associated with possibly non-semisimple
finite unimodular ribbon tensor categories. State spaces are given by
admissible skein modules, and we prescribe the TQFT on handle attachments. We
give some explicit algebraic conditions on the category to define this TQFT,
namely to be "chromatic non-degenerate". As a by-product, we obtain an
invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in
the "twist non-degenerate" case, an invariant of 3-manifolds. Our construction
generalizes the Crane--Yetter--Kauffman TQFTs in the semi-simple case, and the
Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole
construction is very elementary, and we can easily characterize invertibility
of the TQFTs, study their behavior under connected sum and provide some
examples.
