Criegee intermediates are produced in the ozonolysis of unsaturated hydrocarbons in the troposphere, and understanding their fate is a prerequisite to modeling climate-controlling atmospheric aerosol formation. Although some experimental and theoretical rate data are available, they are incomplete and partially inconsistent, and they do not cover the tropospheric temperature range. Here, we report quantum chemical rate constants for the reactions of stabilized formaldehyde oxide (CH2OO) and acetaldehyde oxide (syn-CH3CHOO and anti-CH3CHOO) with H2O and for their unimolecular reactions. Our results are obtained by combining post-CCSD(T) electronic structure benchmarks, validated density functional theory potential energy surfaces, and multipath variational transition state theory with multidimensional tunneling, coupled-torsions anharmonicity, and high-frequency anharmonicity. We consider two different types of reaction mechanisms for the bimolecular reactions, namely, (i) addition-coupled hydrogen transfer and (ii) double hydrogen atom transfer (DHAT). First, we show that the MN15-L exchange-correlation functional has kJ/mol accuracy for the CH2OO + H2O and syn-CH3CHOO + H2O reactions. Then we show that, due to tunneling, the DHAT mechanism is especially important in the syn-CH3CHOO + H2O reaction. We show that the dominant pathways for reactions of Criegee intermediates depend on altitude. The results we obtain eliminate the discrepancy between experiment and theory under those conditions where experimental results are available, and we make predictions for the full range of temperatures and pressures encountered in the troposphere and stratosphere. The present results are an important cog in clarifying the atmospheric fate and oxidation processes of Criegee intermediates, and they also show how theoretical methods can provide reliable rate data for complex atmospheric processes.