Two sided bounds of the geometric convolution with subexponential components are derived. Such a convolution can be viewed as a ruin probability (in risk theory), waiting time distribution (in queueing), a distribution of the 1-st break down time (in reliability), etc. In contrast to other known bounds, these have genuine asymptotic behaviour. They are derived with the help of new characterizations of subexponential distributions. Several examples with widely used subexponential distributions are considered.