  In this paper, we propose a definition of Neron models of arbitrary Deligne
1-motives over Dedekind schemes, extending Neron models of semi-abelian
varieties. The key property of our Neron models is that they satisfy a
generalization of Grothendieck's duality conjecture in SGA 7 when the residue
fields of the base scheme at closed points are perfect. The assumption on the
residue fields is unnecessary for the class of 1-motives with semistable
reduction everywhere. In general, this duality holds after inverting the
residual characteristics. The definition of Neron models involves careful
treatment of ramification of lattice parts and its interaction with
semi-abelian parts. This work is a complement to Grothendieck's philosophy on
Neron models of motives of arbitrary weights.
