Intentionality is a central feature of our understanding of the world. We daily attribute intentional states (like beliefs, desires or perceptual states) to explain the behavior of other agents, and many theories appeal to them to understand more complex notions. Nonetheless, intentional states are puzzling entities. This article explains what intentionality is and why it is so important and problematic at the same time. Secondly, it examines various naturalistic theories, which seek to show that intentionality is compatible with a scientific (...) worldview. Finally, given that all extant proposals face significant difficulties, it explores the available options in case no naturalistic theory can succeed. (shrink)