  Using high-quality observed period-luminosity relations in both Magellanic
Clouds in VIJHKs bands and optical and near-infrared Wesenheit indices we
determine the effect of metallicity on Cepheid P-L relations by comparing the
relative distance between LMC and SMC as determined from the Cepheids to the
distance difference between the Clouds which has been derived with very high
accuracy from late-type eclipsing binary systems. Within an uncertainty of 3%
which is dominated by the uncertainty on the mean metallicity difference
between the Cepheid populations in LMC and SMC we find metallicity effects
smaller than 2% in all bands and in the Wesenheit indices, consistent with a
zero metallicity effect. This result is valid for the metallicity range from
-0.35 dex to -0.75 dex corresponding to the mean [Fe/H] values for classical
Cepheids in LMC and SMC, respectively. Yet most Cepheids in galaxies beyond the
Local Group and located in the less crowded outer regions of these galaxies do
fall into this metallicity regime, making our result important for applications
to determine the distances to spiral galaxies well beyond the Local Group. Our
result supports previous findings which indicated a very small metallicity
effect on the near-infrared absolute magnitudes of classical Cepheids, and
resolves the dispute about the size and sign of the metallicity effect in the
optical spectral range. It also resolves one of the most pressing problems in
the quest towards a measurement of the Hubble constant with an accuracy of 1%
from the Cepheid-supernova Ia method.
