  A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if
it admits a unique additive decomposition in powers of linear forms.
Identifiability is expected to be very rare. In this paper we conclude a work
started more than a century ago and we describe all values of $d$ and $n$ for
which a general polynomial of degree $d$ in $n+1$ variables is identifiable.
This is done by classifying a special class of Cremona transformations of
projective spaces.
