  In this paper, we show that if $X$ is a smooth variety of general type of
dimension $m \geq 2$, for which its canonical map induces a double cover onto
$Y$, where $Y$ is a projective bundle over $\mathbf P^1$, or onto a projective
space or onto a quadric hypersurface, embedded by a complete linear series,
then the general deformation of the canonical morphism of $X$ again is
canonical and again induces a double cover. The second part of the article
deals with the existence or non existence of canonical double structures on
rational varieties. The negative result in this article has consequences for
the moduli of varieties of general type of arbitrary dimension. The results
here show that there is an entire component, that is hyperelliptic in
infinitely many moduli spaces of higher dimensional varieties of general type.
This is in sharp contrast with the case of curves or surfaces of lower Kodaira
dimensions.
