The kinetics and mechanism of the reactions OH + BrO --> products (1) and OD + BrO --> products (2) have been studied in the temperature ranges of 230-355 K and 230-320 K, respectively, and at total pressure of 1 Torr of helium using the discharge-flow mass spectrometric method. The following Arrhenius expressions for the total rate constants have been obtained from the kinetics of BrO consumption in excess of OH(OD) radical: k(1) = (1.65 +/- 0.30) x 10(-11) exp{(250 +/-50)/T} cm(3) molecule(-1) s(-1) (with k(1) = (3.8 +/- 0.9) x 10(-11) cm(3) molecule(-1) s(-1) at T = 298 K) and k(2) = (1.7 +/- 0.6) x 10(-11) exp{(230 +/- 100)/T} cm(3) molecule(-1) s(-1) (with k(2) = (3.7 +/- 0.9) x 10(-11) cm(3) molecule(-1) s(-1) at T= 298 K), where uncertainties are twice the standard deviation. From the kinetics of HBr formation, the upper limit of the rate constant of the reaction OH + BrO --> HBr + O-2 (Ib) has been determined at T = 298 K: k(1b) < 1.0 x 10(-12) cm(3) molecule(-1) s(-1) (k(1b)/k(1) < 0.03 for the branching ratio of channel 1b). Similarly, for the reaction OD + BrO --> DBr + O-2 (2b), the rate constant at T = 298 K has been determined: k(2b) = (3.7 +/- 1.8) x 10(-13) cm(3) molecule(-1) s(-1) (which corresponds to the branching ratio k(2b)/k(2) = (1.0 +/- 0.5) X 10(-2)) In addition, the rate constant of the reaction OD + DO2 --> D2O + O-2 (3) has been measured for the first time: k(3) = (3.8 +/- 0.9) x 10(-11) cm(3) molecule(-1) s(-1) at T = 298 K. This work suggests that the additional HBr source from the OH + BrO reaction, although significant, does not appear to be sufficient to explain the difference between current modeled and observed stratospheric HBr concentrations.