Something like 40% of people over 60 have some sort of memory problems according to an article on Science Daily. In general it seems this is caused by the gradual loss of brain cells and changes in brain chemistry.

The article, Rote Learning Improves Memory In Seniors, describes a study where seniors spent 6 weeks of intensive memory training followed by six weeks without it. At the end of the 12 week period, all of the 25 volunteers were able to learn and remember better than they were at the start of the study.

It does appear that using the mind is one of the best ways to keep it in shape.