Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a nondegenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of two different types of odd geometries on supermanifolds.