  The concept of a CDW induced by Fermi-surface nesting originated from the
Peierls idea of electronic instabilities in purely 1D metals and is now often
applied to charge ordering in real low-dimensional materials. The idea is that
if Fermi surface contours coincide when shifted along the observed CDW wave
vector, then the CDW is considered to be nesting-derived. We show that in most
cases this procedure has no predictive power, since Fermi surfaces either do
not nest at the right wave vector, or nest more strongly at the wrong vector.
We argue that only a tiny fraction, if any, of the observed charge ordering
phase transitions are true analogues of the Peierls instability because
electronic instabilities are easily destroyed by even small deviations from
perfect nesting conditions. Using prototypical CDW materials NbSe$_2$,
TaSe$_2$, and CeTe$_3$, we show that such conditions are hardly ever fulfilled,
and that the CDW phases are actually structural phase transitions, driven by
the concerted action of electronic and ionic subsystems, \textit{i.e.,}
\textbf{q}-dependent electron-phonon coupling plays an indispensable part. We
also show mathematically that the original Peierls construction is so fragile
as to be unlikely to apply to real materials. We argue that no meaningful
distinction between a CDW and an incommensurate lattice transition exists.
