Most of the digital technology around us processes and stores information in a sequential, well-ordered manner. For example, a computer hard drive stores data bits in a structured way that allows them to be retrieved in order. Cell phone systems transmit and receive data as sequences of bits coded to allow the receiver to reconstruct the information in the same order it was sent. Recent technological advances such as DNA sequencing technologies, however, defy this ordered information paradigm. Such technologies generate data consisting of many short, out-of-order fragments. Processing this data is akin to assembling a jigsaw puzzle, where the desired information is only conveyed by the final assembled picture. Developing powerful algorithms for these tasks is important for several applications in the field of genomics and for the development of emerging molecular data storage technologies. The goal of this project is to extend techniques from the ordered digital world - codes, algorithms, and an information-theoretic framework - to these emerging out-of-order settings. This should enable new data storage paradigms to be deployed and lead to the development of new computational methods to analyze genomics data. The project will seek to extend Information Theory techniques to out-of-order information scenarios and to characterize how much information can be reliably conveyed by an unordered set of data fragments. The research will be organized along three thrusts with impor