The influence on fusion of coupling to the breakup process is investigated for reactions where at least one of the colliding nuclei has a sufficiently low binding energy for breakup to become an important process. Elastic scattering, excitation functions for sub- and near-barrier fusion cross sections, and breakup yields are analyzed for 6,7Li+59Co. Continuum-discretized coupled-channels (CDCC) calculations describe well the data at and above the barrier. Elastic scattering with 6Li (as compared to 7Li) indicates the significant role of breakup for weakly bound projectiles. A study of 4,6He induced fusion reactions with a three-body CDCC method for the 6He halo nucleus is presented. The relative importance of breakup and bound-state structure effects on total fusion is discussed