  Normalized correlation functions provide expedient means for determining the
photon-number properties of light. These higher-order moments, also called the
normalized factorial moments of photon number, can be utilized both in the fast
state classification and in-depth state characterization. Further,
non-classicality criteria have been derived based on their properties. Luckily,
the measurement of the normalized higher-order moments is often
loss-independent making their observation with lossy optical setups and
imperfect detectors experimentally appealing. The normalized higher-order
moments can for example be extracted from the photon-number distribution
measured with a true photon-number-resolving detector or accessed directly via
manifold coincidence counting in the spirit of the Hanbury Brown and Twiss
experiment. Alternatively, they can be inferred via homodyne detection. Here,
we provide an overview of different kind of state classification and
characterization tasks that take use of normalized higher-order moments and
consider different aspects in measuring them with free-traveling light.
