In this talk we will give a short review of the Itô stochastic integal and present an extension of this integral onto a class of anticipating stochastic processes introduced by Ayed and Kuo in 2008. Next, we will present some of the properties of the new integral. Among those are the near-martingale property --- an anticipating counterpart of the martingale property, an isometry formula, and several Itô formulas. The results presented in this talk are based on joint work with Hui-Hsiung Kuo and Anuwat Sae-Tang.