We have performed detailed optical measurements of ultrathin InAs/InP quantum wells grown by metalorganic vapor phase epitaxy. Absorption and photoluminescence excitation spectra reveal the excitonic resonances associated with two- and three-monolayer-thick InAs layers. Photoluminescence spectra also show an emission band at intermediate photon energies which is associated with excitons localized in thin InAs quantum dots. Polarization-dependent measurements clearly show the heavy-hole or light-hole nature of the quantum well resonances. Such an identification of both type of transitions provides a test for electronic structure models. We find that the energy positions of the excitonic transitions in ultrathin InAs/InP quantum wells are not consistent with calculations based on the envelope function model.