We study the exciton wave functions and the optical properties of cylindrical molecular aggregates. The cylindrical symmetry allows for a decomposition of the exciton Hamiltonian into a set of effective one-dimensional Hamiltonians, characterized by a transverse wave number k2. These effective Hamiltonians have interactions that are complex if the cylinder exhibits chirality. We propose analytical ansatze for the eigenfunctions of these one-dimensional problems that account for a finite cylinder length, and present a general study of their validity. A profound difference is found between the Hamiltonian for the transverse wave number k2=0 and those with k2 not equal 0. The complex nature of the latter leads to chiral wave functions, which we characterize in detail. We apply our general formalism to the chlorosomes of green bacteria and compare the wave functions as well as linear optical spectra (absorption and dichroism) obtained through our ansätze with those obtained by numerical diagonalization as well as those obtained by imposing periodic boundary conditions in the cylinder's axis direction. It is found that our ansätze, in particular, capture the finite-length effect in the circular dichroism spectrum much better than the solution with periodic boundary conditions. Our ansätze also show that in finite-length cylinders seven superradiant states dominate the linear optical response.