  We show that being a general fibre of a Mori fibre space is a rather
restrictive condition for a Fano variety. More specifically, we obtain two
criteria (one sufficient and one necessary) for a Q-factorial Fano variety with
terminal singularities to be realised as a fibre of a Mori fibre space, which
turn into a characterisation in the rigid case. We apply our criteria to figure
out this property up to dimension three and on rational homogeneous spaces. The
smooth toric case is studied and an interesting connection with K-semistability
is also investigated.
