  We study in this article the dual of a (strictly) commutative group stack $G$
and give some applications. Using the Picard functor and the Picard stack of
$G$, we first give some sufficient conditions for $G$ to be dualizable. Then,
for an algebraic stack $X$ with suitable assumptions, we define an Albanese
morphism $a_X : X\to A^1(X)$ where $A^1(X)$ is a torsor under the dual
commutative group stack $A^0(X)$ of $Pic_{X/S}$. We prove that $a_X$ satisfies
a natural universal property. We give two applications of our Albanese
morphism. On the one hand, we give a geometric description of the elementary
obstruction and of universal torsors (standard tools in the study of rational
varieties over number fields). On the other hand we give some examples of
algebraic stacks that satisfy Grothendieck's section conjecture.
