  This paper discusses two aspects of current research on the Cepheid
period-luminosity (P-L) relation: the derivation of mid-infrared (MIR) P-L
relations and the investigation of multi-phase P-L relations. The MIR P-L
relations for Cepheids are important in the James Webb Space Telescope era for
the distance scale issue, as the relations have potential to derive the Hubble
constant within ~ 2% accuracy - a critical constraint in precision cosmology.
Consequently, we have derived the MIR P-L relations for Cepheids in the Large
and Small Magellanic Clouds, using archival data from Spitzer Space Telescope.
We also compared currently empirical P-L relations for Cepheids in the
Magellanic Clouds to the synthetic MIR P-L relations derived from pulsational
models. For the study of multi-phase P-L relations, we present convincing
evidence that the Cepheid P-L relations in the Magellanic Clouds are highly
dynamic quantities that vary significantly when considered as a function of
pulsational phase. We found that there is a difference in P-L relations as a
function of phase between the Cepheids in each of the Clouds; the most likely
cause for this is the metallicity difference between the two galaxies. We also
investigated the dispersion of the multi-phase P-L relations, and found that
the minimum dispersion do not differ significantly from the mean light P-L
dispersion.
