Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification? 
In particular, suppose $X$ is a singular projective variety and $U$ is the smooth locus. The question is does there always exist a smooth projective variety $Y$ containing $U$? If we add the condition that $U$ is open in $Y$ is $Y$ uniquely determined upto isomorphism?



Non-uniqueness of smooth compactification
The main idea is this: You can always find an $X$ such that $X\supseteq U$ and $X\setminus U\supseteq \Sigma := \mathrm{Sing} X$. Then in characteristic zero apply Hironaka's resolution theorem, which says that there exists a resolution of singularities $\pi:Y\to X$ such that $\pi$ is an isomorphism over $X\setminus \Sigma\supseteq U$. In particular, $\pi^{-1}: U\hookrightarrow Y$ gives an embedding.


In positive characteristic resolution is not known in general and similarly this embedding result is not known either (although I am not saying that knowing this would prove the existence of resolutions).


There is actually a newer, expanded version of Kollár's paper in book form: 
[Lectures on Resolution of Singularities](http://press.princeton.edu/titles/8449.html). 


And of course it is not unique as long as $U$ itself is not projective, since as you can always blow-up $Y$ outside of $U$.


