Peto's Paradox was introduced by the UK epidemiologist Richard Peto in 1977. It is still generally considered unsolved.

Simply stated : All animal cells have the potential to turn cancerous. Large animals have many more cells, therefore larger animals should be subject to higher rates of cancer. But they're not. In mammals for example, the larger animals in general have broadly similar rates to the smaller ones.

And there are strong outliers - whales, for example, only rarely get cancer, but it's relatively common in cats and mice.

In humans, the current rate for 'Western' societies mean that each individual has around a 50% chance of developing it at some stage.

A suggested 'solution' to the paradox was published in 2015 - which pointed out that larger organisms tend to have bigger and more slowly-dividing cells with lower energy turnover, which may reduce the risk of developing cancer.

[…] we believe that although Peto's paradox is likely a multifactor phenomenon, its solution will remain elusive unless cell size and metabolic rate are taken into consideration.

Source : Evol Appl. 2015 Jan; 8(1): 2–8

It's well established that larger animals do tend to have slower metabolisms - and that faster metabolisms can lead to increased cancer rates (ref.) - but it's also evident that the larger species tend to live considerably longer, and thus will presumably encounter many more opportunities (timewise) for chance mutations caused by carcinogenic substances and/or radiation etc. etc..

The 'multifactor' aspect to which the paper above refers includes the possibility that genetic differences (genes which make cancer less likely - and others which make it more likely) could be important. At present though, there's no solid understanding as to why. For example, cancer is the leading cause of death in dogs, whereas in sheep it's relatively rare.

Whatever factors are operating, they must exert a strong effect, because, in the case of mammals for instance, some of the larger examples have many thousands of times more cells than the smaller ones.

Also see : Oncogenesplugin-autotooltip__plain plugin-autotooltip_bigOncogenes

Several hundred oncogenes - genes which which have to potential to increase the risk of cancer - have now been identified. The way(s) in which they do so is as yet unknown.

"A fundamental aim of cancer research is to identify the molecular changes that cause normal cells to evolve into malignant tumors. Malignant transformation has been proposed to occur as a consequence of the accumulation of genomic aberrations that successively overcome the cellular barriers to malignancy. and Naked mole-rat - cancer resistanceplugin-autotooltip__plain plugin-autotooltip_bigNaked mole-rat - cancer resistance

"Naked mole-rats (NMRs; Heterocephalus glaber) are highly adapted, eusocial rodents renowned for their extreme longevity and resistance to cancer. Because cancer has not been formally described in this species, NMRs have been increasingly utilized as an animal model in aging and cancer research."