Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$? 



Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$?
This is a notorious open problem. For the moment the simplest *compact* four-manifold that is announced to admit (infinite number of) exotic smooth structures is $S^2\times S^2$. This result is contained here : <http://arxiv.org/abs/1005.3346>


I have to say that I am not at all an expert in the area 
(also it seems that the above paper is not yet published). On the other hand there are several published papers showing that $CP^2\sharp 3\overline{CP^2}$ admit exotic smooth structures.


Also, it might be worth to recall that by a theorem of Yau a complex surface homeomrophic to $CP^2$ always has the standard smooth structure (in other words $CP^2$ admits a unique holomorphic structure up to bi-homolorphism). While for $S^2\times S^2$ this is still unknown (is there a surface of general type homeomorphic to $S^2\times S^2$?) 


