The random motion of nearly spherical bubbles in the turbulent flow behind a grid is studied experimentally. In quiescent water these bubbles rise at high Reynolds number. The turbulence is generated by an active grid of the design of Makita (1991), and can have turbulence Reynolds number R[lambda] of up to 200. Minor changes in the geometry of the grid and in its mode of operation improves the isotropy of the turbulence, compared with that reported by Makita (1991) and Mydlarski & Warhaft (1996). The trajectory of each bubble is measured with high spatial and temporal resolution with a specially developed technique that makes use of a position-sensitive detector. Bubble statistics such as the mean rise velocity and the root-mean-square velocity fluctuations are obtained by ensemble averaging over many identical bubbles. The resulting bubble mean rise velocity is significantly reduced (up to 35%) compared with the quiescent conditions. The vertical bubble velocity fluctuations are found to be non-Gaussian, whereas the horizontal displacements are Gaussian for all times. The diffusivity of bubbles is considerably less than that of fluid particles. These findings are qualitatively consistent with results obtained through theoretical analysis and numerical simulations by Spelt & Biesheuvel (1997).