The Turaev-Viro 6j-symbol invariant is a real number associated to any compact 3-manifold, and is a very strong topological invariant, related to quantum groups and to Jones-Witten invariants (for "empty" manifolds), whose construction involves a lot of sophisticated and difficult mathematics. A very simple version (using only some classical 3-dimensional geometric topology due to Castler and Matveev, which will be described in the talk) of this invariant will be presented.