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Let $K$ be the function field of a smooth projective curve $X$ over ahigher-dimensional local field $k$. We define Tate-Shafarevich groups of acommutative group scheme via cohomology classes locally trivial at eachcompletion of $K$ coming from a closed point of $X$. We establish dualitytheorems between Tate-Shafarevich groups for finite groups schemes, for tori,for groups of multiplicative type, and even for 2-term complexes of tori. Weapply these results to the weak approximation for tori over $K$ and to thestudy of the obstruction to the local-global principle for $K$-torsors under aconnected linear algebraic group. We also give examples and counter-examples tothe local-global principle for central simple algebras over $K$.  Soit $K$ le corps des fonctions d'une courbe projective lisse X sur un corpslocal sup\'erieur $k$. On d\'efinit les groupes de Tate-Shafarevich d'unsch\'ema en groupes commutatif en consid\'erant les classes de cohomologie quideviennent triviales sur chaque compl\'et\'e de $K$ provenant d'un pointferm\'e de $X$. On \'etablit des th\'eor\`emes de dualit\'e arithm\'etiqueentre des groupes de Tate-Shafarevich pour les modules finis, pour les tores,pour les groupes de type multiplicatif, et m\^eme pour les complexes \`a deuxtermes de tores. On applique ces r\'esultats \`a l'approximation faible pourles tores sur $K$ et \`a l'\'etude du principe local-global pour les$K$-torseurs sous un groupe lin\'eaire connexe. On exhibe aussi des exemples etdes contre-exemples au principe local-global pour les alg\`ebres simplescentrales sur $K$.

Th\'eor\`emes de dualit\'e pour les corps de fonctions sur des corps locaux sup\'erieurs et applications arithm\'etiques