  For a number field $K$, an algebraic variety $X/K$ is said to have the
Hilbert Property if $X(K)$ is not thin. We are going to describe some examples
of algebraic varieties, for which the Hilbert Property is a new result.
  The first class of examples is that of smooth cubic hypersurfaces with a
$K$-rational point in $\mathbb{P}_n/K$, for $n \geq 3$. These fall in the class
of unirational varieties, for which the Hilbert Property was conjectured by
Colliot-Th\'el\`ene and Sansuc.
  We then provide a sufficient condition for which a surface endowed with
multiple elliptic fibrations has the Hilbert Property. As an application, we
prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.
