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."

Source : Veterinary Pathology , Volume: 53 issue: 3, page(s): 691-696

The paper cited above described two cases of pre-cancerous lesions in zoo-housed Naked Mole-Rats (NMR). This was the first time that any cancer, of any type, had been found in the species. Making them (almost*) the only mammal which is not routinely susceptible to the disease.

Because of their innate resistance to cancer, NMRs are the subject of intense research attempting to discover why.

There are currently several theories. One focuses on mammalian genes p16 and p27, which, it has been suggested, act as a double barrier to uncontrolled cell proliferation. Another is based on the fact that skin cells of the naked mole-rat have high levels of HMW-HA - a natural sugary substance that is said to prevent tumours developing. A third theory is that NMRs have has a unique fragmented ribosomal RNA structure which enables more accurate protein translation than other mammals.

See: Wikipedia

*Note : The Golan Heights blind mole-rat (Spalax golani) and the Judean Mountains blind mole-rat (Spalax judaei) are also extremely resistant to cancer - but apparently via a different mechanism. See : PNAS November 20, 2012 109 (47) 19392-19396

Also see :Peto's paradoxplugin-autotooltip__plain plugin-autotooltip_bigPeto's paradox

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.