Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.


A well known theorem independently discovered by Eilenberg and Watts states that every $k$-linear, right exact, cocontinuous functor $F: A\mathsf{-Mod}\to B\mathsf{-Mod}$ is of the form $M\otimes\_A-$ for some bimodule ${}\_B M\_A$ (in fact $M=F(A)$).


A theorem of Rickard (with some refinements by other people like Keller) also states that if there is an equivalence $D^b(A) \to D^b(B)$ there is also an equivalence of the form $X\otimes\_A^\mathbb{L}-$ for some $X\in D^b(B\otimes\_k A^{op})$.


A reasonable question to ask is then:



> 
> Is every equivalence $D^b(A) \to D^b(B)$ itself of the form $X\otimes\_A^{\mathbb{L}}-$ for some bounded complex of bimodules $X$?
> 
> 
> 


or more generally:



> 
> Is every exact functor $D^b(A) \to D^b(B)$ of triangulated categories which commutes with all direct sums (plus maybe some other reasonable condition) of the form $X\otimes\_A^\mathbb{L}-$ for some bounded complex of bimodules $X$ ?
> 
> 
> 



Eilenberg-Watts theorem for the derived category
Your first question seems to be a famous open question due to Rickard, see <https://academic.oup.com/jlms/article-abstract/s2-43/1/37/888935?redirectedFrom=PDF> . 


It is known in several special cases, such as for hereditary algebras, see <http://home.ustc.edu.cn/~xwchen/Personal%20Papers/A%20note%20on%20standard%20equivalence.pdf> . 


A very recent approach can be found here: <https://www.sciencedirect.com/science/article/pii/S000187081830207X> .


