The term 'Maya', in Indian traditions, refers to our sensory perception of the world and, as such, to a superficial reality (or 'un-reality') that we must look beyond to find the inner reality of things. Applied to the study of language, we perceive sounds, a superficial reality, and then we seek structures, the underlying reality in what we call phonology, morphology, and syntax. This volume starts with an introduction by the editors, which shows how the various papers contained in the volume reflect the spectrum of research interests of Andrea Calabrese, as well as his influence on the work of colleagues and his students. Contributors, united in their search for the abstract structures that underlie the appearances of languages include linguists such as Adriana Belletti, Paola Benincà, Jonathan Bobaljik, Gugliemo Cinque, David Embick, Mirko Grimaldi, Harry van der Hulst, Michael Kenstowicz, Maria Rita Manzini, Andrew Nevins, Elizabeth Pyatt, Luigi Rizzi, Leonardo Savoia, Laura Vanelli, Bert Vaux, Susi Wurmbrand, as well as a few junior researchers including Mariachiara Berizzi, Giuliano Bocci, Stefano Canalis, Silvio Cruschina, Irina Monich, Beata Moskal, Diego Pescarini, Joseph Perry, Roberto Petrosino, and Kobey Schwayder.