  We construct Abel maps for a stable curve $X$. Namely, for each one-parameter
deformation of $X$ with regular total space, and every integer $d>0$, we
construct by specialization a map $\alpha^d_X$ from the smooth locus of $X^d$
to the moduli scheme of balanced line bundles on semistable curves over $X$.
For $d=1$, we show that $\alpha^1_X$ naturally extends over $X$, and does not
depend on the choice of the deformation; we give a precise description of when
it is injective.
