TITLE: Compact variety which is not projective
QUESTION [5 upvotes]: While reading Andreas Gathmann's notes on Algebraic Geometry, I stumbled upon this statement: "Projective varieties form a large class of “compact” varieties that do admit such a unified global description. In fact, the class of projective varieties is so large that it is not easy to construct a variety that is not (an open subset of) a projective variety.".
I know that we can sometimes glue affine varieties together and create compact spaces (in fact, Gathmann constructs $\mathbb{P^1}(\mathbb{C})$ as a compactification of $\mathbb{A}^1$). Also affine varieties are not compact unless they are single points. But my question is: is there an example of a variety which is "compact" but not projective? 
Gathman does not provide such an example, so maybe someone here can help.

REPLY [4 votes]: Such example does not exist in dimension 1. For dimension 3, see the appendix B in Hartshorne, Example 3.4.1
As Liu pointed out below, there is a list discussing related questions in Hartshorne Chapter II Beneath Remark 4.10.2.