  The versatility of renewal theory is owed to its abstract formulation.
Renewals can be interpreted as steps of a random walk, switching events in
two-state models, domain crossings of a random motion, etc. We here discuss a
renewal process in which successive events are separated by scale-free waiting
time periods. Among other ubiquitous long time properties, this process
exhibits aging: events counted initially in a time interval [0,t] statistically
strongly differ from those observed at later times [t_a,t_a+t]. In complex,
disordered media, processes with scale-free waiting times play a particularly
prominent role. We set up a unified analytical foundation for such anomalous
dynamics by discussing in detail the distribution of the aging renewal process.
We analyze its half-discrete, half-continuous nature and study its aging time
evolution. These results are readily used to discuss a scale-free anomalous
diffusion process, the continuous time random walk. By this we not only shed
light on the profound origins of its characteristic features, such as weak
ergodicity breaking. Along the way, we also add an extended discussion on aging
effects. In particular, we find that the aging behavior of time and ensemble
averages is conceptually very distinct, but their time scaling is identical at
high ages. Finally, we show how more complex motion models are readily
constructed on the basis of aging renewal dynamics.
