The patio around my pool has shown some cracks recently, and today a piece about as big as my fist fell off. What is the best way to repair the broken piece, and what kind of preventative maintenance can I take to maintain the life of the patio?
There are epoxy products that can be used to repair stone, such as this one
This will work to glue the seams of the big chunks back in place. While it could be used to fill other breaks or gaps, it probably will not look very attractive.
Also, as DA01 says in his comment, the overhang makes this a very vulnerable area. Usually stones which overhang have a thicker dimension.
There is not too much you can do unless you want to put a facing under the overhang to help support it, a fairly big job involving cutting strips of matching stone. You could watch for the beginnings of cracks near the edges and fill them with an outdoor epoxy based material. This will reduce the chance that water gets in them and freezes, which leads to major cracking.
You could also tread lightly on the edges.