We study the molecular gas properties of high-$z$ galaxies observed in the ALMA Spectroscopic Survey (ASPECS) that targets a $sim1$ arcmin$^2$ region in the Hubble Ultra Deep Field (UDF), a blind survey of CO emission (tracing molecular gas) in the 3mm and 1mm bands. Of a total of 1302 galaxies in the field, 56 have spectroscopic redshifts and correspondingly well-defined physical properties. Among these, 11 have infrared luminosities $L_{rm{}IR}>10^{11}$ L$_odot$, i.e. a detection in CO emission was expected. Out these, 7 are detected at various significance in CO, and 4 are undetected in CO emission. In the CO-detected sources, we find CO excitation conditions that are lower than typically found in starburst/SMG/QSO environments. We use the CO luminosities (including limits for non-detections) to derive molecular gas masses. We discuss our findings in context of previous molecular gas observations at high redshift (star-formation law, gas depletion times, gas fractions): The CO-detected galaxies in the UDF tend to reside on the low-$L_{rm{}IR}$ envelope of the scatter in the $L_{rm{}IR}-L_{rm{}CO}$ relation, but exceptions exist. For the CO-detected sources, we find an average depletion time of $sim$ 1 Gyr, with significant scatter. The average molecular-to-stellar mass ratio ($M_{rm{}H2}$/$M_*$) is consistent with earlier measurements of main sequence galaxies at these redshifts, and again shows large variations among sources. In some cases, we also measure dust continuum emission. On average, the dust-based estimates of the molecular gas are a factor $sim$2-5$times$ smaller than those based on CO. Accounting for detections as well as non-detections, we find large diversity in the molecular gas properties of the high-redshift galaxies covered by ASPECS.
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