What I know is that Bombieri proved that for algebraic surfaces $h^0(2K_X)\not=0$
and the map defined by $|nK_X|$ is always birational for every $n\geq 5$.
Moreover if you take a minimal surface $X$ with geometric genus 2 and $(K_X)^2=1$, then $|4K_X|$ is not birational.

The same questions are very interesting and widely open in higher dimension. 

We know only partial results for 3-folds and 4-folds, you can see for example the introduction of [http://uk.arxiv.org/PS_cache/arxiv/pdf/1001/1001.3340v1.pdf][1] if you are interested.



  [1]: http://uk.arxiv.org/PS_cache/arxiv/pdf/1001/1001.3340v1.pdf