Can an algebraic variety be embedded in projective space?

Question

Let $X$ be an algebraic variety over field $k$ .
$X$ can be embedded in a complete variety by Nagata's compactification theorem.

Moreover, can we embed $X$ in a projective space $\mathbb{P}_k^n$ $???$
Please give me references which contain either proof or counter examples.

Answer 1

From KReiser's comment:

Hironaka's example (link at Wikipedia) shows that arbitrary varieties over fields cannot necessarily be embedded in projective space.