Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 2/A, 1-62 (1987).

The paper is devoted to the geometric aspects of the theory of symplectic groupoids and algebroids, which was inspired by certain problems in symplectic mechanics. Roughly speaking, a Lie groupoid

${\Gamma}$, whose manifold of units is denoted by

${{\Gamma}}_{0}$, is a smooth groupoid in the sense of Ehresmann and a Lie algebroid, which is a vector bundle over

${{\Gamma}}_{0}$ with a bracket operation, is an “infinitesimal version” of a Lie groupoid. A Lie groupoid

${\Gamma}$ endowed with a symplectic 2- form is called a symplectic groupoid, if the graph of its multiplication is a lagrangian submanifold in

$(-{\Gamma})\times {\Gamma}\times {\Gamma}$. On the other hand, every Poisson manifold

$({{\Gamma}}_{0},{{\Lambda}}_{0})$ induces a Lie algebroid structure on

${T}^{*}{{\Gamma}}_{0}\to {{\Gamma}}_{0}$ and every Lie algebroid of such a type is called a symplectic algebroid. The main result of the paper reads that a Lie algebroid is the algebroid of a local symplectic groupoid if and only it it is a symplectic algebroid. This assertion has several concrete applications in symplectic geometry.