First measurements of the probability density function of geometrical pathlengths (PDF-GP) for the skylight transmitted from clear and cloudy skies to the terrestrial surface are reported. The measurements are performed using a novel technique, which includes spectroscopically highly resolving observations of the O2 A-band (760--780 nm) and a Laplacian back transformation of the intensity ratios measured for the set of individual O2 A-band rotational absorption lines. The technique provides a new and powerful tool to study the radiative transfer (RT) of the atmosphere for different kinds of atmospheric aerosol loadings and cloudiness. First measurements show, as expected, that the photons transmitted by clouds to ground generally experience longer geometrical paths inside the cloud than expected for a corresponding clear sky condition. The comparison of the measured oxygen A-band absorption system with plane-parallel discrete ordinate radiative transfer (DISORT) model calculations shows a reasonable agreement for direct Sun and clear sky zenith observations. For cloudy sky the same comparison shows a good agreement between the observed and the modeled weak absorption lines but systematically larger observed than modeled optical densities for strong absorption lines. Our data also suggest that this significant discrepancy is most likely caused by shortcomings in the physical description of the RT by nonstatistical models, which do not properly account for the fractal nature of the terrestrial cloud cover. In particular our study reveals that the photon path statistics involved with the multiple Mie scattering inside clouds is not properly modeled using the DISORT code. Closing this gap in our understanding of the cloud sky RT may possibly provide much of the explanation for the frequently lower modeled than measured cloudy sky absorption of the solar radiation. ¿ 1998 American Geophysical Union