  We present results from the Large Magellanic Cloud Near-infrared Synoptic
Survey (LMCNISS) for classical and type II Cepheid variables that were
identified by the Optical Gravitational Lensing Experiment (OGLE-III)
catalogue. Multiwavelength time-series data for classical Cepheid variables are
used to study light-curve structures as a function of period and wavelength. We
exploit a sample of $\sim$1400 classical and $\sim$80 type II Cepheid variables
to derive Period--Wesenheit relations that combine both optical and
near-infrared data. The new Period--Luminosity and Wesenheit relations are used
to estimate distances to several Local Group galaxies (using classical
Cepheids) and to Galactic globular clusters (using type II Cepheids). By
appealing to a statistical framework, we find that fundamental-mode classical
Cepheid Period--Luminosity relations are non-linear around 10--18 days at
optical and near-IR wavelengths. We also suggest that a non-linear relation
provides a better constraint on the Cepheid Period--Luminosity relation in type
Ia Supernovae host galaxies, though it has a negligible effect on the
systematic uncertainties affecting the local measurement of the Hubble
constant.
