The theory of the optical properties of linear and tubular molecular J-aggregates is reviewed. The primary optical excitations in these systems are Frenkel excitons, which may be delocalized over many molecules. The collective nature of these excitations gives rise to special optical properties and dynamics, which are of interest for purely scientific reasons, but also enable the application of J-aggregates as photographic sensitizers and artificial light-harvesting systems. The focus of this paper is on the effect of aggregate geometry, disorder, and temperature on the absorption and fluorescence spectra. Also transport of excitations between J-aggregates is discussed. Connection is made to experiments on aggregates of cyanine dyes and natural light-harvesting systems.