A sample of 37 Galactic , 10 LMC and 6 SMC cepheids is compiled for which individual metallicity estimates exist and BVIK photometry in almost all cases .
The Galactic cepheids all have an individual distance estimate available .
For the MC objects different sources of photometry are combined to obtain improved periods and mean magnitudes .
A multi-parameter Period-Luminosity relation is fitted to the data which also solves for the distance to the LMC and SMC .
When all three galaxies are considered , without metallicity effect , a significant quadratic term in \log P is found , as previously observed and also predicted in some theoretical calculations .
For the present sample it is empirically determined that for \log P < 1.65 
linear PL -relations may be adopted , but this restricts the sample to only 4 LMC and 1 SMC cepheid .
Considering the Galactic sample a metallicity effect is found in the zero point in the VIWK PL -relation ( -0.6 \pm 0.4 or -0.8 \pm 0.3 mag/dex depending on the in- or exclusion of one object ) , in the sense that metal-rich cepheids are brighter .
The small significance is mostly due to the fact that the Galactic sample spans a narrow metallicity range .
The error is to a significant part due to the error in the metallicity determinations and not to the error in the fit .
Including the 5 MC cepheids broadens the observed metallicity range and a metallity effect of about -0.27 \pm 0.08 mag/dex in the zero point is found in VIWK , in agreement with some previous empirical estimates , but now derived using direct metallicity determinations for the cepheids themselves .