Understanding the nature and the structure of human language coincides with capturing the constraints which make a conceivable language possible or, equivalently, with discovering whether there can be any impossible languages at all. This book explores these related issues, paralleling the effort of a biologist who attempts at describing the class of impossible animals. In biology, one can appeal for example to physical laws of nature (such as entropy or gravity) but when it comes to language the path becomes intricate and difficult for the physical laws cannot be exploited. In linguistics, in fact, there are two distinct empirical domains to explore: on the one hand, the formal domain of syntax, where different languages are compared trying to understand how much they can differ; on the other, the neurobiological domain, where the flow of information through the complex neural networks and the electric code exploited by neurons is uncovered and measured. By referring to the most advanced experiments in Neurolinguistics the book in fact offers an updated descriptions of modern linguistics and allows the reader to formulate new and surprising questions. Moreover, since syntax - the capacity to generate novel structures (sentences) by recombining a finite set of elements (words) - is the fingerprint of all and only human languages this books ultimately deals with the fundamental questions which characterize the search for our origins.