All right, first post, so please be nice. 
UVB and UVA are both carcinogenic, although they do work differently. If you are exposed to large amounts of these during adolescence, you have an increased chance of getting cancer. However, the vast majority of people get cancer around 60.
My question is, what’s up with the delay? Why do you get exposed to a mutagen and, 40 years later, get the tumours going? Does the cancer, lurk in your bloodstream, waiting for the right time?
Only for some weird definition of “vast majority”:
Actual data shows skin cancer rates rise steadily with age.
For many cancers, there is a presumed sequence of development whereby more and more genetic mutations accumulate over time. Simply put, the more time that’s passed, the more mutations will accumulate. Eventually, it’s believed that a critical threshold of mutations is passed and the cancer ‘begins’.
As to why there are ‘more and more’ mutations as time goes on, in part it’s simply statistics - the older you are, the more cell divisions there’ve been, and hence the more chances there have been for mutations to occur. Likewise, the older you are, the more (cumulative) exposure there’s been to natural causes of mutations such as radiation from sunlight and cosmic rays, from the earth’s interior, naturally occurring carcinogens, etc.
Further, there is presumably a gap between when the cancer “begins” and when the doctor sits you down and says “I have some bad news…”
While the answers given make sense, in particular the accumulation of damage over the years, I wonder if the op might also end up having a different answer.
Both aging and cancer are related to cellular senescence. It may be that understanding the mechanisms involved in normal aging will provide better understanding of cancer and that there is overlap between the processes - not restricted to the accumulation of mutations and damage over the years.
That said I do not believe that understanding yet exists.
Quickly searching some one area I can find of overlap is focused on the mitochondria.
One, in relation to aging (“Aging: The Biology of Senescence”):
Two, in relation to cancer (“Mitochondrial mutations in cancer”):
So it may be that both normal aging and an increasing predisposition to cancer with age are the result of common processes.
It is certainly clear that oxidative damage accumulates as we age. That’s been known for years. Whether it’s due to leaky mitochondria, or decreased activity of ROS scavenging enzymes, or some other mechanism isn’t entirely clear. But these molecules can and do damage DNA, amongst other molecules, and that damage builds up as we get olde.
One thing I have wondered is why animals like mice and rats are so cancer-prone when they don’t live that long; there simply isn’t much time for mutations to accumulate. Even when you consider their rate of aging, they surely have fewer cell divisions than a human over their lifetime.
That is a very interesting point Michael63129. In fact mice and humans both get cancer at the same fraction of their maximal lifespan:
So cancer and aging do not accumulate linearly with time, but do co-vary. It therefore follows that if both are a result of accumulated insults/damage (oxidative or otherwise) then they both depend at least as much on DNA repair/protection mechanisms that vary greatly between species of different lifespans. Or the insult/damage model is not the whole story and alternatively there are mechanisms that have been evolutionarily selected to occur that cause an individual past reproductive age (and past additional benefit greater than cost to subsequent gene copies being passed on) that limit lifespan, cause aging, and increase cancer risk.
That’s really interesting.
There have been experiments with mice I believe where they were put on starvation diets to prolong their lives (I’d rather die young, thanks) - do you know whether they tested those mice for incidents of cancer, and whether the timing of those matched the baseline mouse lifespan, or the longer skinnymouse lifespan?
I did not know but a quick search shows this:
and this
(Please note the same did not hold true in the recent primate models. But that is another subject.)
Erghk. Thanks guys. Biology is so complicated… 
Sorry if my response at least was less than clear.
Your specific example though illustrates why there has to more to it than exclusively the accumulation of damage over time - melanoma has a peak of new cases in the 60s (as you stated and as illustrated in the link provided by naita - but the risk for melanoma is mainly determined by exposure early in life. The oft quoted stat is that one blistering sunburn in childhood doubles melanoma risk compared to five blistering sunburns during adulthood to double the risk.
The damage is mostly done early but does not come out until later. I am guessing that we have pretty good detection and control systems that keep those potentially cancerous cells in check, and that eliminate any of those cells that begin to get out of control, but that begin to lose efficacy as we age … i.e. not as much that there is so much more cancer cell producing damage occurring as that our ability to monitor/contain the results of that damage deteriorates with age - resulting both in the aging process itself and in increased cancer risk. And that loss of monitoring/containment ability is to some large degree intrinsic to any particular species.
On the other hand, a smaller body size within a species, including humans (why do women live longer and have lower rates of CHD? Mostly because they are smaller than men, according to this paper, which also suggests that caloric restriction increases lifespan due to a smaller body size), is highly correlated to a reduced risk of chronic diseases, including cancer, possibly because a smaller body requires less upkeep (cell divisions, repair).
Sunburn is an interesting subject. I picked up the tidbit below form Sam Kean’s book The Violinist’s Thumb
UV radiation damages the double helix shaped DNA strands in a very specific manner. Wherever there are two thymine bases side by side, UV radiation fuses them together at an angle. This kinks up the DNA strand causing redness and burning. If the strand gets really kinked it will break at points. The DNA is good at self-repair if the break is on one side of the double helix strand. When both sides of the strand break the repair is less precise, especially if there are a lot of repairs from a severe sunburn. In these instances base pairs can be replaced incorrectly or deleted ( shortening the strand), causing mutations.
These mutations are not always cancerous but they do accumulate over time.
This is also interesting in terms of excessive sun exposure causing early aging at the cellular level. Telomeres are the non-coding repetitve strands that mark the end of each chromosone. Research into aging has shown that the telomeres shorten with age – ( this is a BIG problem with cloning, animals cloned from adult cells have shorter telomeres which decreases life-span ) The repetitive sequence that makes up the telomeres includes 2 adjacent Thymine pairs … the human telomere sequence is TTAGGG. Damage to the T bases in telomeres can result in them being shortened during repair, causing the cell to age faster
Thank you very much, that was quite interesting!
Those statistics were from the UK. It is widely known that the UK hasn’t seen a sunny day since 1973.
According to my sources you can get a tan from standing in the English rain. If it can give you a tan it ought to have similar cancer-causing effects as sunlight. 
The possibilities that the emergence of cancer with aging is
a) to some degree a consequence of the same normal programming, possibly evolutionarily selected for programming, that causes normal aging
and
b) a consequence to a great degree of diminished function (with age) in how systems identify and eliminate misbehaving always extant precancerous cells, keeping their numbers in check and eliminating the most dangerous ones before they turn into full blown cancer cells and spread, more so perhaps than even how many of those precancerous cells there are
suggests to me a somewhat different focus for future research than has, to my non-expert eye, than has been emphasized to date.
I wonder what work has been done on those areas?
I think aging and cancer is very much an active area of research. I helped out on a study many years ago looking at sort of the flip side, which was figuring out the cause of childhood cancers. Childhood cancers tend to be very different from adult cancers, and there’s not a whole lot of overlap between them. I tested some samples of heel-stick blood from newborns who later developed childhood lymphomas. It turned out that many of them had been born with one of the characteristic genetic defects associated with this type of lymphoma. This suggested that perhaps there’s a period during pregnancy and development where the infant genome is particularly susceptible to at least some types of chromosomal rearrangements. They can easily pick up one of the mutations required for cancer, making it more likely that they’ll develop it down the line.
Anyway, the point is that researchers have been paying attention to the age factor for a good long time.