In this paper the algebraic form of quadratic symplectic mappings in \(\mathbb{R}^{2n}\) with the standard symplectic structure is investigated. For the planar case this leads to the familiar Hénon maps. For \(n \geq 2\) it is shown that any quadratic map can be written as product of a shear map to be multiplied from the right and left by affine symplectic maps. As a consequence it is shown that the inverse map of a quadratic symplectic map is also quadratic. In the generic case normal forms of such quadratic maps are derived, where special attention is given to the 4-dimensional case. Also generalized symplectic maps are considered and the question of strange attractors for such maps is raised. In an aside the stability of elliptic fixed points for area-preserving quadratic mappings in the plane is discussed where Feldman's theorem in the theory of transcendental numbers is used.