Moral realism, also known as moral objectivism or objective morality, says that moral facts exist. There are two forms which this takes: ethical naturalism and ethical non-naturalism. Moral absolutism is often a form moral realist theories take, but not always.
This view claims moral facts rest on natural things, such as happiness, pleasure, wellbeing etc.
In contrast, this theory claims moral facts have no natural correlates, but are sui generis, abstract objects somewhat like Platonic forms.
Not only moral anti-realists but ethical non-naturalists have attacked this view. Ethical non-naturalists usually follow G. E. Moore, who said it commits a naturalistic fallacy to claim natural properties are good. They say it is a fallacy as we can always ask why (for example) pleasure embodies "the good" (the open question argument). Similar to this is the Is-ought problem. Ethical naturalists however respond that while pleasure, say, is not logically equivalent to the good, examination shows it is.
To begin with, many question the existence of any abstract objects at all. For people who reject the idea, obviously non-naturalism is a failure. Even apart from this, the "argument from queerness" has also been made from a naturalist perspective, saying moral facts of the kind they espouse would be highly queer, as they don't correspond with anything natural, and thus probably don't exist.
Perhaps the main reason both of these views have been rejected is called the "argument from disagreement". Put simply, if moral facts exist, there is a difficulty with explaining why such frequent, profound disagreement occurs on moral issues. Moral realists, however, note that disagreement exists in many areas (science included) yet that does not mean there are no facts to be found.