In this work we present a new theoretical approach for diffusion-controlled reactions which generalizes the classic theory of Smoluchowski and that of Collins and Kimball. In this approach, the non-Markovian effect is taken into account in the framework of a generalized diffusion equation with a time-dependent diffusion coefficient. When Smoluchowski’s absorbing boundary condition is considered, we have found the exact analytical solution of the generalized diffusion equation despite the presence of a time-dependent diffusion coefficient. Our generalized Smoluchowski theory removes the unphysical singularity in reaction rate, k(t), at t=0 in the classic Smoluchowski theory. This allows to evaluate the initial value of reaction rate. The result shows that the initial reaction rate is overestimated for collision-induced reactions. This is due to the inappropriateness of the absorbing boundary condition for describing the collision-induced reactions. To take into account more properly these kinds of reactions, Collins-Kimball’s boundary condition is considered. In this case, a perturbation method and an approximation ansatz are developed to find an approximate solution, The approximation for reaction rate contains an adjustable parameter. A scheme is prescribed for the optimal choice of this parameter in short and long time regions. With the help of this scheme, we obtain an approximation of the reaction rate which has the virtue to give simultaneously the exact asymptotic behavior at long time and the right initial value of the reaction rate.