  Fano models - consisting of a Hamiltonian with discrete-continuous spectrum -
are one of the basic toy models in spectroscopy. They have been succesfull in
explaining the lineshape of experiments in atomic physics and condensed matter.
These models however have largely been out of the scope of dissipative
dynamics, with ony a handful of works considering the effect of a thermal bath.
Yet in nanostructures and condensed matter systems, dissipation strongly
modulates the dynamics. In this article, we present an overview of the
theoretical works dealing with Fano interferences coupled to a thermal bath and
compare them to the scattering formalism. We provide the solution to any
discrete-continuous Hamiltonian structure within the wideband approximation
coupled to a Markovian bath. In doing so, we update the toy models that have
been available for unitary evolution since the 1960s. We find that the Fano
lineshape is preserved as long as we allow a rescaling of the parameters, and
an additional Lorentzian contribution that reflects the destruction of the
interference by dephasings. We discuss the pertinence of each approach -
dissipative and unitary - to different experimental setups: scattering,
transport and spectroscopy of dissipative systems. We finish by discussing the
current limitations of the theories due to the wideband approximation and the
memory effects of the bath.
