Theories about who Neanderthals were?

What are some theories about who Neanderthals were? What is their relationship to modern humans? What do genetic studies tell us about these questions?

Together with an Asian people known as Denisovans, Neanderthals are our closest ancient human relatives. Scientific evidence suggests our two species shared a common ancestor. Current evidence from both fossils and DNA suggests that Neanderthal and modern human lineages separated at least 500,000 years ago.