  Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for
studying solvable quantum potentials. It is the simplest method to build
Hamiltonians with prescribed spectra in the spectral design. The key is to pair
two Hamiltonians through a finite order differential operator. Some related
subjects can be simply analyzed, as the algebras ruling both Hamiltonians and
the associated coherent states. The technique has been applied also to periodic
potentials, where the spectra consist of allowed and forbidden energy bands. In
addition, a link with non-linear second order differential equations, and the
possibility of generating some solutions, can be explored. Recent applications
concern the study of Dirac electrons in graphene placed either in electric or
magnetic fields, and the analysis of optical systems whose relevant equations
are the same as those of SUSY QM. These issues will be reviewed briefly in this
paper, trying to identify the most important subjects explored currently in the
literature.
