Cepheids have been used as distance indicators for almost a century. Through their well defined period–luminosity relation, their distance can be found from their period of pulsation alone. However, in recent years, an important uncertainty has arisen in its calibration, namely the effect of metallicity on the period–luminosity relation. It is important that we quantify this effect as Cepheids are used to tie down the base of the extragalactic distance scale. Without an accurate calibration for the nearest distances, the distances we measure for all other objects will be subject to systematic uncertainties. This thesis studies how the chemical composition of a Cepheid affects the zero-point of the period–luminosity relation, and quantifies the resulting change in measured distance modulus. The first study uses Cepheids in two areas of M33; a sample from the metal–rich central region is compared with the more metal–poor sample from the southern spiral arm. Period–luminosity relations in the reddening–free Wesenheit index Wvi are constructed for the two samples and the measured distance moduli are compared. A significant offset is found between the two samples. Effects such as reddening and blending are ruled out, leaving the change in metallicity as the only possible explanation for the discrepancy. The second study presented covers the whole of M33. Here, a sample of around 600 Cepheids is used, and the effect of metallicity on Wgi and Wri is measured. The same conclusion is found; the change in composition significantly affects the measured distance modulus of the Cepheid and the period–luminosity relation must be changed to take this into account. In addition, the Sloan band period–luminosity relations are derived empirically for the first time. The metallicity gradient of M33 is also assessed. It is found that the slope must be steep, at least in the central few kpc of the galaxy, but may flatten off at larger radial distances.