Higher-order hadronic contributions to the anomalous magnetic moment
of the electron, muon, and $\tau$ lepton are considered in
detail. As a main result we find a reduction by $-18 \times 10^{-11}$
for the g-2 of the muon as compared to previous calculations.
Analytical expressions for the kernel functions of higher-order
hadronic effects are presented. We employ the method of asymptotic
expansions to calculate kernel functions analytically
in terms of a series expansion in the lepton mass.