At first the author proves the existence of a unique measurable solution for the following random variational inequality: Find for every elementary event

$\omega $ an element

$\widehat{y}\in \mathcal{K}$ depending on

$\omega $ such that

$\beta (\omega ,\widehat{y},z-\widehat{y})\ge \lambda (\omega ,z-\widehat{y})$,

$z\in \mathcal{K}$, where

$\mathcal{K}$ is a closed convex non-empty subset of a separable Hilbert space and for every

$\omega $,

$\lambda (\omega ,\xb7)$ and

$\beta (\omega ,\xb7,\xb7)$ being a linear form, respectively, bilinear form. It is assumed that

$\beta $ is nonnegative. Then the author considers in detail a more specialized inequality, where the data decompose in deterministic data and given real-valued random variables. Existence and uniqueness results are given, various approximation procedures and its convergence are studied. The general theory is applied to a Helmholtz-like elliptic equation with Signorini boundary conditions.