Invariant Measures for Stochastic Nonlinear Schrödinger Equations : Numerical Approximations and Symplectic Structures This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations including the well posedness ergodicity symplecticity and multi symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations introduces geometric structures including symplecticity and (conformal) multi symplecticity for nonlinear Schrödinger equations and their numerical approximations and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations. This book will appeal to researchers who are interested in numerical analysis stochastic analysis ergodic theory partial differential equation theory etc Invariant measures and ergodicity Invariant measures for stochastic differential equations Invariant measures for stochastic nonlinear Schrödinger equations Geometric structures and numerical schemes for nonlinear Schrödinger equations Numerical invariant measures for damped stochastic nonlinear Schrödinger equations Approximation of ergodic limit for conservative stochastic nonlinear Schrödinger equations