  Correlation functions play an important role for the theoretical and
experimental characterization of many-body systems. In solid-state systems,
they are usually determined through scattering experiments whereas in
cold-gases systems, time-of-flight and in-situ absorption imaging are the
standard observation techniques. However, none of these methods allow the
in-situ detection of spatially resolved correlation functions at the
single-particle level. Here we give a more detailed account of recent advances
in the detection of correlation functions using in-situ fluorescence imaging of
ultracold bosonic atoms in an optical lattice. This method yields single-site
and single-atom-resolved images of the lattice gas in a single experimental
run, thus gaining direct access to fluctuations in the many-body system. As a
consequence, the detection of correlation functions between an arbitrary set of
lattice sites is possible. This enables not only the detection of two-site
correlation functions but also the evaluation of non-local correlations, which
originate from an extended region of the system and are used for the
characterization of quantum phases that do not possess (quasi-)long-range order
in the traditional sense.
