Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov $n$-widths of compact sets from some classes of functions have been evaluated. In conclusion we adduce some numerical illustrations for 2-D Volterra equations.