Absolute rate coefficients for the reaction of hydroxyl radicals (HO) with methanol, HO + CH3OH --> products (R1), and with ethanol, HO + C2H5OH --> products (R2) were measured over a range of temperatures using pulsed laser photolytic generation of HO coupled to its time resolved detection by pulsed laser induced fluorescence. The accuracy of the rate constants obtained was enhanced by on-line optical absorption measurements of the alcohol concentration. The temperature dependence of the rate coefficients is given by: k1(210-351 K) = 6.67 x 10(-18) T2 exp(140/T) cm3 molecule(-1) s(-1) with a rate coefficient at room temperature of (9.3 +/- 0.7) x 10(-13) cm3 molecule(-1) s(-1). For k2 we obtained: k2(216-368 K) = 4.0 x 10(-12) exp(-42/T) and a room temperature rate coefficient of (3.35 +/- 0.17) x 10(-12) cm3 molecule(-1) s(-1). The total error (at 95% confidence) associated with the rate coefficients derived from the expressions describing the temperature dependence is estimated as 7% at all temperatures. The present results, which extend the database on these reactions to cover temperatures relevant for the upper troposphere, are compared to previously published measurements, and values of k1 and k2 are recommended for atmospheric modelling.