This paper settles the existence issue of non-central Wishart distributions,
a research topic initiated by Wishart (1928), and with important contributions
by e.g., L\'evy (1937), Gindikin (1975), Shanbhag (1988), Das Peddada and
Richards (1991). We present a new method involving the theory of affine Markov
processes, which allows a full characterization of non-central Wishart
distributions by joint conditions on shape and non-centrally parameter. In
particular, it is not sufficient anymore that the shape parameter belongs to
the Gindikin ensemble, as is in the central case.