The Ito theory of stochastic integration deals with adapted stochastic processes. We introduce the class of instantly independent stochastic processes as the counterpart of the Ito theory. A new stochastic integral is defined for integrands involving both adapted and instantly independent stochastic processes. In the definition of the new stochastic integral, the crucial idea of forming Riemann-like sums is the evaluation points for the integrands, i.e., the adapted factors are evaluated at the left endpoints of subintervals, while the instantly independent factors are evaluated at the right endpoints. The new stochastic integral has many advantages, e.g., the multiple Ito-Wiener integrals can be treated as iterated stochastic integral in the same way as in ordinary calculus. We will present recent results concerning this new stochastic integral such as the Ito isometry, Ito's formula, and solutions of stochastic differential equations with anticipating initial conditions.