  We review recent progresses in the study of flat band systems, especially
focusing on the fundamental physics related to the singularity of the flat
band's Bloch wave functions. We first explain that the flat bands can be
classified into two classes: singular and nonsingular flat bands, based on the
presence or absence of the singularity in the flat band's Bloch wave functions.
The singularity is generated by the band crossing of the flat band with another
dispersive band. In the singular flat band, one can find special kind of
eigenmodes, called the non-contractible loop states and the robust boundary
modes, which exhibit nontrivial real space topology. Then, we review the
experimental realization of these topological eigenmodes of the flat band in
the photonic lattices. While the singularity of the flat band is topologically
trivial, we show that the maximum quantum distance around the singularity is a
bulk invariant representing the strength of the singularity which protects the
robust boundary modes. Finally, we discuss how the maximum quantum distance or
the strength of the singularity manifests itself in the anomalous Landau level
spreading of the singular flat band when it has a quadratic band-crossing with
another band.
