  As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such
that through $n\geq 2$ general points there passes an irreducible curve $C$ of
degree $\delta\geq n-1$ we have $N\leq \pi(r,n,\delta+r(n-1)+2)$, where
$\pi(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus
of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ of
degree $d$. A lot of examples of varieties as in the title and attaining the
previous bound for the embedding dimension are constructed from Castelnuovo
varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is
also proved that all extremal varieties as above are of this kind, except
possibly when $n>2$, $r>1$ and $\delta=2n-3$.
  One of the main results of the paper is the classification of extremal
varieties $X^{r+1}\subset \mathbb P^{2r+3}$ 3-covered by twisted cubics and not
of Castelnuovo type. Interesting examples are provided by the so called {\it
twisted cubics over complex Jordan algebras of rank 3}, as pointed out by
Mukai. By relating to an extremal variety 3-covered by twisted cubics, via
tangential projection, a quadro-quadric Cremona transformation in $\mathbb P^r$
we are able to classify all these object either for $r\leq 4$ or under the
smoothness assumption. In the last case we obtain that they are either smooth
rational normal scrolls (hence of Castelnuovo type) or the Segre embeddings of
$\p^1\times Q^r$ or one of the four Lagrangian Grassmannians. We end by
discussing some open problems pointing towards the equivalence of these
apparently unrelated objects: extremal varieties 3-covered by twisted cubics,
quadro-quadric Cremona transformations of $\mathbb P^r$ and complex Jordan
algebras of dimension $r+1$ and of rank three.
