  This is the final version of the 2007 preprint titled "On the derived
category of 1-motives, I". It has been substantially expanded to contain a
motivic proof of (two thirds of) Deligne's conjecture on 1-motives with
rational coefficients, hence the new title. Compared to the 2007 preprint, the
additions mainly concern an abstract theory of realisations with weight
filtrations; Deligne's conjecture is tackled though them by an adjunction game.
