Normal projective varieties of degree 5

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$(k) \geq 0$.