There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these two constructions? (This question is related to, and indeed the inspiration for, one of my previous [questions](https://math.stackexchange.com/questions/141/model-category-structures-on-categories-of-complexes-in-abelian-categories).)



Derived categories and homotopy categories
Yes. The former is a special case of the latter. There is a model category structure on the category of (say bounded) chain complexes of objects in your given abelian category. The weak equivalences are the quasi-isomorphisms, and the homotopy category is the derived category. 


In the case of R-modules, for a ring R, this is explained in detail in
[this paper by Dwyer-Spalinski](http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf).


