  We list the irreducible reduced and not degenerate normal projective
varieties $X\subset\mathbb{P}^N$ of dimension $n$ and degree five defined over
an algebraically closed field $k$ of char$(k) = 0$. In the smooth case, or when
$n = 2$, we give also classification results for any algebraically closed field
$k$ of char$(k) \geq 0$.
