If $A,B$ are objects of an abelian category $\mathcal{A}$ and $n \in \mathbb{N}$, there is a very nice and useful description of $\mathrm{Ext}^n(A,B)$. Namely, it is just the set of morphisms $A \to B[n]$ in the derived category $D(\mathcal{A})$. For example, this gives more elegant formulations of the Yoneda product $\mathrm{Ext}^n(A,B) \otimes \mathrm{Ext}^m(B,C) \to \mathrm{Ext}^{n+m}(A,C)$ and Serre duality on a smooth projective scheme $X$.
Now let $\mathcal{A}$ be an abelian $\otimes$-category with enough projectives / flat objects. Is there a similar description of $\mathrm{Tor}_n(A,B)$ using the derived category? More precisely, can we manipulate $A,B$ inside of $D(\mathcal{A})$ (using shifts, Homs, tensor products etc.) to get the abelian group $\mathrm{Tor}_n(A,B)$ without talking about projective resolutions?