I believe that Konrad Lorenz was the first one to advance the evolutionary theory of cuteness. Stephen Jay Gould wrote an article about it (pdf) using Mickey Mouse as an example (don't be dissuaded by the author's identity). Lorenz argued that we respond with awwwws and nurturing behavior to features that distinguish infant humans from adults, like large round heads, large eyes, small pudgy limbs, and clumsy movements, even if they belong to another animal or a nonliving thing.

There has been research on why animals are cute, again going back to Lorenz, and I think it's generally accepted that the young of many species are cute to us because they share cute-inducing features that are common to that developmental stage, some species (like pandas) seem cute just because out of the wide variety of species some of them happen to have cute-inducing features, and some species (like dogs) seem cute because humans have bred them that way. I'm not sure if there is research on whether adults of other species find their own young to be cute.

Human infants are unusually helpless for an unusually long period of time, which helps explain why humans are so attuned to cuteness (and why there would be a bias towards over-identifying instances of cuteness, which evolutionarily is the less costly error). That doesn't explain why bunnies are cuter than babies, though, or why non-humans dominate the top of our cutest list. Perhaps they just have more of the cute-inducing features. Humans occupy a small portion of body-space, and if you move from the region occupied by adult humans to the region occupied by human infants and then go even further along the same dimensions, you could run into regions occupied by other animals. But why would this happen for cuteness but not sexiness?