Neutral atoms in highly-excited (Rydberg) states are strongly-polarisable particles, which due to their exaggerated properties can experience quantum effects and interactions over macroscopic distances. Many-body systems of Rydberg atoms offer unique opportunities to create and investigate strong correlations in ultra-cold atomic samples. This work will be devoted to studying the collective effects that arise in the excitation of a dense cold gas towards Rydberg states. Many-body effects caused by Rydberg enhanced interactions can manifest themselves in dramatic ways, such as the generation of entanglement, structure formation, and exotic quantum phases.A new experimental system for studying strongly correlated quantum matter is available for this project. We will investigate the emergence of crystalline order in the excited Rydberg gas due to interactions. A new method to directly image the Rydberg atoms will be developed for this purpose. Finally, the interactions properties of Rydberg atoms will be mapped onto the degenerate, long-lived ground state bath. This will extend the field of ultra-cold gases to the study of new strongly-correlated many-body phases of matter, shedding new light on self-ordering mechanisms and ultimately creating a powerful new platform for quantum information science. It will raise the applicant's international recognition, and establish him inside Europe as a leading researcher in this new and quickly evolving field.