We present a detailed investigation of the Cepheid distance scale by using both theory and observations .
Through the use of pulsation models for fundamental mode Cepheids , we found that the slope of the Period-Luminosity ( P - L ) relation covering the entire period range ( 0.40 \leq log P \leq 2.0 ) becomes steeper when moving from optical to near-infrared ( NIR ) bands , and that the metallicity dependence of the slope decreases from the B - to the K -band .
The sign of the metallicity dependence for the slopes of the P - L _ { V } and P - L _ { I } relation is at odds with some recent empirical estimates .

We determined new homogeneous estimates of V - and I -band slopes for 87 independent Cepheid data sets belonging to 48 external galaxies with nebular oxygen abundance 7.5 \leq 12+ \log ( O/H ) \leq 8.9 .
By using Cepheid samples including more than 20 Cepheids , the \chi ^ { 2 } test indicates that the hypothesis of a steepening of the P - L _ { V,I } relations with increased metal content can be discarded at the 99 % level .
On the contrary , the observed slopes agree with the metallicity trend predicted by pulsation models , i.e .
the slope is roughly constant for galaxies with 12+ \log ( O/H ) < 8.17 and becomes shallower in the metal-rich regime , with a confidence level of 62 % and 92 % , respectively .
The \chi ^ { 2 } test concerning the hypothesis that the slope does not depend on metallicity gives confidence levels either similar ( PL _ { V } , 62 % ) or smaller ( PL _ { I } , 67 % ) .

We investigated the dependence of the Period-Wesenheit ( P - W ) relations on the metal content and we found that the slopes of optical and NIR P - W relations in external galaxies are similar to the slopes of Large Magellanic Cloud ( LMC ) Cepheids .
They also agree with the theoretical predictions suggesting that the slopes of the P - W relations are independent of the metal content .
On this ground , the P - W relations provide a robust method to determine distance moduli relative to the LMC , but theory and observations indicate that the metallicity dependence of the zero-point in the different passbands has to be taken into account .

To constrain this effect , we compared the independent set of galaxy distances provided by Rizzi et al. ( 2007 ) using the Tip of the Red Giant Branch ( TRGB ) with our homogeneous set of extragalactic Cepheid distances based on the P - W relations .
We found that the metallicity correction on distances based on the P - WBV relation is \gamma _ { B,V } = -0.52 mag dex ^ { -1 } , whereas it is vanishing for the distances based on the P - { WVI } and on the P - { WJK } relations .
These findings fully support Cepheid theoretical predictions .