Not all of the cells in your body replace their populations on a regular basis. Some groups of cells will last almost an entire lifetime, and across that lifetime they will become increasingly damaged and dysfunctional. Here is an overview of that process: "It is now generally accepted that aging and eventual death of multicellular organisms is to a large extent related to macromolecular damage by mitochondrially produced reactive oxygen species, mostly affecting long-lived postmitotic cells, such as neurons and cardiac myocytes. ... The inherent inability of autophagy and other cellular degradation mechanisms to completely remove damaged structures results in the progressive accumulation of garbage, including cytosolic protein aggregates, defective mitochondria and lipofuscin - an intralysosomal indigestible material. ... The slow accumulation of lipofuscin within lysosomes seems to depress autophagy, resulting in reduced mitochondrial turnover. The latter are not only functionally deficient but also produce increased amounts of reactive oxygen species, prompting [the creation of lipofuscin]. Moreover, defective and enlarged mitochondria are [only poorly recycled] and constitute a growing population of badly functioning organelles ... The progress of these changes seems to result in enhanced oxidative stress, decreased ATP production, and collapse of the cellular catabolic machinery, which eventually is incompatible with survival." This is the garbage catastrophe of the cell: garbage builds up, which makes things worse and leads to more garbage. Eventually it kills the cell, and enough of this will kill you too.

Link: http://www.ncbi.nlm.nih.gov/pubmed/19650712