  The collisions of three-dimensional bright solitary matter waves formed from
atomic Bose-Einstein condensates are shown to exhibit rich behaviour.
Collisions range from being elastic to completely destructive due to the onset
of collapse during the interaction. Through a detailed quantitative analysis we
map out the role of relative phase, impact speed and interaction strength. In
particular, we identify the importance of the collapse time in the onset of
unstable collisions and show how the relative phase controls a population
transfer between the waves. Our analysis enables us to interpret recent
experimental observations of bright solitary matter waves.
