We mammals just can't regenerate as well as lower animals - but we all evolved from the same ancestors, so the suspicion is that we might retain at least some of the necessary mechanisms to regrow organs and limbs, just buried and inactive. Some studies have uncovered possible hints of this: you might recall the gene engineered MRL mice that have superior regenerative abilities due to inactivation of p21, for example.
Here, researchers note the discovery of superior natural regenerative abilities in a rodent species - which should hopefully lead to some further insight into how we might make humans regenerate more capably:
Two species of African spiny mouse have been caught at something no other mammal is known to do - completely regenerating damaged tissue. ... Acomys kempi and Acomys percivali [have] skin that is brittle and easily torn, which helps them to escape predators by jettisoning patches of their skin when caught or bitten. ... whereas normal laboratory mice (Mus musculus) grow scar tissue when their skin is removed, African spiny mice can regrow complete suites of hair follicles, skin, sweat glands, fur and even cartilage.
Tissue regeneration has not been seen in mammals before, but it is common in crustaceans, insects, reptiles and amphibians. Some lizards can regrow only their tails, whereas some salamanders can regenerate entire limbs, complete with bones and muscle.
The researchers say that their next step will be to work out the molecular mechanisms and genetic circuits that direct the regeneration process. It's unlikely that these mice have evolved an entirely new method of regrowing tissue ... Rather, the genes that direct regeneration in salamanders are probably switched off in mammals, but have been switched back on in African spiny mice. ... By looking at the common genetic blueprints that exist across vertebrates, we hope to find the ones that we could activate in humans. We just need to figure out how to dial the process in mammals back to do something the entire system already knows how to do.