  This work provides an introduction and overview on some basic mathematical
aspects of the single-flux Aharonov-Bohm Schr\"odinger operator. The whole
family of admissible self-adjoint realizations is characterized by means of
four different methods: von Neumann theory, boundary triplets, quadratic forms
and Kre{\u\i}n's resolvent formalism. The relation between the different
parametrizations thus obtained is explored, comparing the asymptotic behavior
of functions in the corresponding operator domains close to the flux
singularity. Special attention is devoted to those self-adjoint realizations
which preserve the same rotational symmetry and homogeneity under dilations of
the basic differential operator. The spectral and scattering properties of all
the Hamiltonian operators are finally described.
