  For two prototype systems, we calculate the exact exchange-correlation
kernels $f_{\mathrm{xc}}(x,x',\omega)$ of time-dependent density functional
theory. $f_{\mathrm{xc}}$, the key quantity for optical absorption spectra of
electronic systems, is normally subject to uncontrolled approximation. We find
that, up to the first excitation energy, the exact $f_{\mathrm{xc}}$ has weak
frequency-dependence and a simple, though non-local, spatial form. For higher
excitations, the spatial behavior and frequency-dependence become more complex.
The accuracy of the underlying exchange-correlation potential is of crucial
importance.
