Simulations predict that galaxies grow primarily through the accretion of gas that has not gone through an accretion shock near the virial radius and that this cold gas flows towards the central galaxy along dense filaments and streams. There is, however, little observational evidence for the existence of these cold flows. We use a large, cosmological, hydrodynamical simulation that has been post-processed with radiative transfer to study the contribution of cold flows to the observed z=3 column density distribution of neutral hydrogen, which our simulation reproduces. We find that nearly all of the HI absorption arises in gas that has remained colder than 10^5.5 K, at least while it was extragalactic. In addition, the majority of the HI is rapidly falling towards a nearby galaxy, with non-negligible contributions from outflowing and static gas. Above a column density of N_HI = 10^17 cm^-2, most of the absorbers reside inside haloes, but the interstellar medium only dominates for N_HI > 10^21 cm^-2. Haloes with total mass below 10^10 Msun dominate the absorption for 10^17<N_HI < 10^21 cm^-2, but the average halo mass increases sharply for higher column densities. Although very little of the HI in absorbers with N_HI <~ 10^20 cm^-2 resides inside galaxies, systems with N_HI > 10^17 cm^-2 are closely related to star formation: most of their HI either will become part of the interstellar medium before z=2 or has been ejected from a galaxy at z>3. Cold accretion flows are critical for the success of our simulation in reproducing the observed rate of incidence of damped Lyman-alpha and particularly that of Lyman limit systems. We therefore conclude that cold accretion flows exist and have already been detected in the form of high column density HI absorbers.
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