Mutations in mitochondrial DNA (mtDNA) are at the center of the mitochondrial free radical theory of aging. Every cell has its herd of bacteria-like mitochondria, toiling to generate chemical energy stores and emitting reactive free radicals as a result of this activity. Each mitochondrion has its own copy of mitochondrial DNA, separate from the DNA in the nucleus of the cell. If mutations change or remove any one of a dozen or so of the most vital genes in a mitochondrion then it will malfunction in ways that can evade the cell's quality control mechanisms. That mitochondrion will divide to create more broken copies, and ultimately all the mitochondria in that cell and all its descendants will become defective. This creates malfunctioning, abnormal cells that export streams of harmful, reactive molecules into the surrounding tissues. This process is one of the root causes of aging, and is why you'll see a fair number of posts here on the repair of mitochondrial DNA as a part of any future toolkit of rejuvenation therapies.
There is some debate over the different types of mutational damage in mitochondrial DNA and their significance, however. Damage ranges all the way from comparatively minor point mutations, in which a single base is substituted, all the way up to catastrophic double strand breaks that require major intervention to restore correctly. To pick one example from past research, scientists have shown that mice loaded up with many more mitochondrial point mutations than usual don't seem to suffer for it. So are point mutations unimportant in mitochondrial function and we should focus more on deletions, in which breaks are poorly repaired or replication failed? Perhaps.
Here is a recent open access paper on this topic. These researchers speculate that it's not damage from free radicals at fault, but rather replication issues as mitochondria reproduce inside the cell. This is an interesting challenge to the prevailing paradigm, but the researchers would then have to explain how genetic changes that alter the levels of free radicals produced by mitochondrial can shift life span so readily in laboratory species. Where is the connection to replication rate and efficiency there? Again, it is worth noting that these researchers are predominantly looking at point mutations (and transitions, a form of point mutation) - and this might not be where the action is in the case of mitochondrial DNA and aging.
Owing to their evolutionary history, mitochondria harbor independently replicating genomes. Failure to faithfully transmit the genetic information of mtDNA during replication can lead to the production of dysfunctional electron transport proteins and a subsequent decline in energy production. Cellularly-derived reactive oxygen species (ROS) and environmental agents preferentially damage mtDNA compared to nuclear DNA. However, little is known about the consequences of mtDNA damage for mutagenesis. This lack of knowledge stems, in part, from an absence of methods capable of accurately detecting these mutations throughout the mitochondrial genome.
Using a new, highly sensitive DNA sequencing strategy, we find that the frequency of point mutations is 10-100-fold lower than what has been previously reported using less precise means. Moreover, the frequency increases 5-fold over an 80 year lifespan. We also find that it is predominantly transition mutations, rather than mutations commonly associated with oxidative damage to mtDNA, that increase with age. This finding is inconsistent with free radical theories of aging.
The bottom line for the prospective longevity engineer is that the outcome of this sort of debate is probably moot. Mitochondrial DNA is getting damaged, this is a definite, well-established difference between old tissue and young tissue, and the prospective mechanisms for repair or replacement of mitochondrial DNA will revert the entirety of that difference regardless of how important or unimportant different forms of mutation happen to be. So the order of the day is to carry on building mitochondrial repair therapies: at this point it would be faster to create them and try them out to settle any debate over effectiveness than to do more research and measurement of mitochondrial mutation types and rates.