In this talk we will give a short review of the new stochastic integral introduced by Ayed and Kuo in 2008. Then, we will present some of the properties of the new integral that were established since its introduction. Among the properties discussed will be an isometry formula, several It\^o formulas, and SDEs with anticipating initial conditions. This talk is based on joint work with Hui-Hsiung Kuo and Anuwat Sae-Tang.