  The theoretical description and the properties of Frenkel excitons in
non-covalently bonded molecular aggregates are reviewed from a multi-exciton
perspective of dissipative quantum dynamics. First, the photophysical and
quantum chemical characterization of the monomeric dye building blocks is
discussed, including the important aspect of electron-vibrational coupling
within the Huang-Rhys model. Supplementing the model by the Coulombic
interactions between monomers, the description of aggregates in terms of
excitonic or vibrational-excitonic bands follows. Besides of giving rise to
complex absorption and emission line shapes, exciton-vibrational interaction is
responsible for energy and phase relaxation and thereby limits the size of
coherent excitations in larger aggregates. Throughout, emphasis is put on the
electronic three-level model as a minimum requirement to describe nonlinear
spectroscopies including effects of two-exciton states such as excited state
absorption and exciton-exciton annihilation. The experimentally observed
characteristics of stationary absorption and fluorescence spectra of aggregates
as well as their temperature dependence are discussed. Examples for ultrafast
spectroscopic experiments including pump-probe studies, photon echo and
two-dimensional spectroscopy are presented and results on the size of coherence
domains and on intra- and interband relaxation are given. Finally, experimental
signatures for exciton-exciton annihilation and their analysis with respect to
the mobility of excitons are described.
