One could not imagine the vast progress made in signal and image processing in the past 50 years without the central contribution of data models. A model imposes a structure on the data, enabling numerous applications. Due to their importance, a considerable research attention has been devoted to the design and use of signal models. Through the past several decades, an evolution of contributions led to a series of constantly improving modeling ideas, and better performance in applications as a consequence. In that respect, the past decade has been certainly the era of sparse and redundant representations, a popular and highly effective model for describing signals.Despite the huge attractiveness and success that this and other signal models have had so far, this field is still at its infancy, with many unanswered questions and major shortcomings, all pointing to unexplored avenues of future research. The overall objective of this proposal is to bring sparsity-based signal modeling to new frontiers by revolutionizing the way these models are defined and practiced.More specifically, this proposal outlines several key research directions that will enable us to overcome existing modeling flaws. These include a thorough investigation of the co-sparse analysis model, one of the next fascinating phases of the field of sparse and redundant representations. This new model suggests an alternative rational and has the potential to outperform earlier models. Other directions to be explored in this project are a super-model built as a tree-constellation of sparsity-based models in an attempt to carve better the signal space, a migration from a union-of-subspaces to a union-of-sets, a systematic study of modeling errors in general, and more. The advances that we aim to make will have a marked impact and open the way towards the next generation of signal models and their use in practice.