  By considering non-trivial global deformations of the Witt (and the Virasoro)
algebra given by geometric constructions it is shown that, despite their
infinitesimal and formal rigidity, they are globally not rigid. This shows the
need of a clear indication of the type of deformations considered. The families
appearing are constructed as families of algebras of Krichever-Novikov type.
They show up in a natural way in the global operator approach to the
quantization of two-dimensional conformal field theory. In addition, a proof of
the infinitesimal and formal rigidity of the Witt algebra is presented.
