  We introduce symplectic left Leibniz algebras and symplectic right Leibniz
algebras as generalizations of symplectic Lie algebras. These algebras possess
a left symmetric product and are Lie-admissible. We describe completely
symmetric Leibniz algebras that are symplectic as both left and right Leibniz
algebras. Additionally, we show that symplectic left or right Leibniz algebras
can be constructed from a symplectic Lie algebra and a vector space through a
method that combines the double extension process and the $T^*$-extension. This
approach allows us to generate a broad class of examples.
