  We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} =
{\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by
${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it a {\it Segre-Veronese variety}
and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for
3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the
expected dimension except for a few cases. A list of examples of defective
$X^s$'s is given.
