We study diffusion in lattices of arbitrary dimensions with a power-law distribution of waiting times \ensuremath{\tau}, P(\ensuremath{\tau})\ensuremath{\sim}${\ensuremath{\tau}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}2}$, \ensuremath{\alpha}1, \ensuremath{\tau}\ensuremath{\ge}1. Using general scaling arguments we find that the asymptotic behavior of the mean-square displacement of a random walker is given , where d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$=${d}_{w}$ for \ensuremath{\alpha}0 and d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$=${d}_{w}${1+${d}_{s}$\ensuremath{\alpha}/[2(1-\ensuremath{\alpha})]} for 0\ensuremath{\le}\ensuremath{\alpha}1 and ${d}_{s}$\ensuremath{\le}2. Here ${d}_{w}$ is the (conventional) diffusion exponent for constant waiting times and ${d}_{s}$ is the fracton dimension of the substrate. Our expression for d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$ is general and holds for Euclidean lattices as well as for random and deterministic fractals. We have also investigated scaling properties of the distribution function P\ifmmode \tilde{}\else \~{}\fi{}(l,t) and the corresponding moments 〈${l}^{q}$〉, where l is the chemical distance the walker traveled in time t. To test our theoretical expressions we have performed extensive computer simulations on the incipient percolation cluster in d=2, using the exact enumeration method. The numerical results agree well with the theoretical predictions.