Is tilting theory extended also to arbitrary derived categories? I was reading papers by Rickard ("Morita theory for derived categories") and Keller ("Derived categories and tilting") on tilting theory in derived categories, they seem to focus mostly on module categories (derived categories D(**Mod-**$A$) for some ring or algebra $A$), I was wondering: Is tilting theory extended also to arbitrary derived categories?

Yes. Wikipedia links to this work in the derived category of an arbitrary abelian category (assumably one such that the derived category exists, e.g. Grothendieck,) although it's not a complete generalization as it uses some infinite direct sums.