Autophagy is a fairly recently-discovered mechanism (compared to, say, apoptosis), but it already seems to be a universally-used pathway, for everything from killing invading bacteria, to clearing plaques and aggregates of misfolded proteins (such as those that cause neurodegenerative diseases), to self-cannibalizing unnecessary cell components for energy and building blocks during starvation conditions.
There's some evidence that autophagy is protective against a wide range of neurodegenerative diseases, so a good bit of research is focused on how to stimulate autophagy in neurons in a safe and controlled fashion (since obviously too much autophagy will kill the cell outright).
What are the most interesting developments in research?
Here's a recent (published Jan 2012) paper claiming to show that exercise triggers autophagy in muscles, and that this autophagy is an important part of how excercise makes you healthier. http://www.nature.com/nature/journal/vaop/ncurrent/full/natu...
However, that paper says nothing about neurons, yet it seems to the basis of most of the "Could excercise stimulate neuroprotective autophagy?" speculation that I can find via Google. So I don't know that there's any published link between excercise and neuronal autophagy. I suspect that it's just the paper above showing a link between excercise and muscular autophagy, combined with the untested hypothesis of "Maybe excercise stimulates autophagy in other tisues too."