  We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the
Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension
of their higher secant varieties. Via inverse systems (apolarity) and the study
of certain spaces of forms we are able, in several cases, to determine whether
those secant varieties are defective or not.
