  We construct Q-factorial terminal Fano varieties, starting in dimension 4,
whose nef cone jumps when the variety is deformed. It follows that de Fernex
and Hacon's results on deformations of 3-dimensional Fanos are optimal. The
examples are based on the existence of high-dimensional flips which deform to
isomorphisms, generalizing the Mukai flop.
  We also extend de Fernex and Hacon's positive results on deformations of Fano
varieties. Toric Fano varieties which are smooth in codimension 2 and
Q-factorial in codimension 3 are rigid. The divisor class group is
deformation-invariant for klt Fanos which are smooth in codimension 2 and
Q-factorial in codimension 3. The Cox ring deforms in a flat family when a
terminal Fano which is Q-factorial in codimension 3 is deformed.
  A side result which seems to be new is that the divisor class group of a klt
Fano variety maps isomorphically to ordinary homology.
