Low-energy electron collisions with the HBr and DBr molecules are addressed from the experimental and theoretical points of view. Relative differential cross sections for the excitation of vibrational levels of HBr and DBr up to v = 6 have been measured as a function of the incident electron energy in the range 0 - 4 eV. In addition to the shape resonance near 2 eV collision energy, intense and narrow threshold peaks are found for the excitation of the v = 1 level of HBr and the v = 1 and v =2 levels of DBr. Measurements with high resolution for rotationally cooled molecules have revealed the existence of sharp oscillatory structures in the elastic and v = 0->1 cross sections in a narrow range below the dissociative-attachment threshold. The dissociative-attachment cross section has been measured with high resolution of the incident electrons in the range 0.2-1.4 eV. The theoretical analysis is based on an improved nonlocal resonance model, which has been constructed on the basis of existing fixed-nuclei electron-HBr scattering phase shifts and accurate ab initio calculations of the bound part of the HBr- potential-energy function. This purely ab-initio-based model is used to calculate integral electron-scattering and dissociative-attachment cross sections for HBr and DBr. The theoretical cross sections agree very well with the experimental data. The observed threshold peaks and Wigner cusp structures in the vibrational excitation functions are correctly reproduced. The sharp structures in the v = 0->0 and v = 0->1 cross sections below the dissociative-attachment threshold, consisting of a superposition of boomerang-type oscillations and quasi-bound levels of the outer well of the HBr- potential-energy function, are quantitatively described by the theory. The high degree of agreement between experiment and theory indicates that the essentials of low-energy electron-HBr collision dynamics are completely understood.