Motivating the category of chain complexes
Chain complexes (of, say, abelian groups) are fundamental in homological algebra and algebraic geometry. For example, using the derived category of an abelian category is very natural nowadays. From another point of view, a lot of people are working on generalizing this ideas to the non abelian world (using simplicial stuff), see for example this answer.
So my question is: why chain complexes? I already know that they are extremely useful, but I'm looking for some theoretical reason for study them.