  The cross-Wigner distribution $W(f,g)$ of two functions or temperate
distributions $f,g$ is a fundamental tool in quantum mechanics and in signal
analysis. Usually, in applications in time-frequency analysis $f$ and $g$
belong to some modulation space and it is important to know which modulation
spaces $W(f,g)$ belongs to. Although several particular sufficient conditions
have been appeared in this connection, the general problem remains open. In the
present paper we solve completely this issue by providing the full range of
modulation spaces in which the continuity of the cross-Wigner distribution
$W(f,g)$ holds, as a function of $f,g$. The case of weighted modulation spaces
is also considered. The consequences of our results are manifold: new bounds
for the short-time Fourier transform and the ambiguity function, boundedness
results for pseudodifferential (in particular, localization) operators and
properties of the Cohen class.
