Our project is devoted to the mathematical analysis of systems of nonlinear partial differential equations of the reaction-diffusion type which serve as the canonical model of the evolution of the density of several populations under the combined effects of reaction and diffusion. Such systems have a rich structure and often exhibit stable propagating solutions, or traveling waves. They have received much interest since the early 30's and nevertheless still present an active field of research. Furthermore, they typically arise in a wide range of applications from both physical and natural sciences. Therefore, the analytical study of such systems is often essential in the understanding of spreading phenomena, e.g. flame combustion, epidemics, ecological or bacterial invasions, spreading of neural activity. The typical questions that have been addressed by mathematicians and of crucial importance in applications can be formulated as follows. Suppose that at initially one is given an initial density of n species, what can be said about the asymptotic behavior of each density as time goes to infinity? Does each species spread and colonize the whole environment, and if yes, can one quantify this spreading? What is the final geometric configuration of the densities, if it exists? For scalar equations having a comparison principle or for monotone systems, answers to the above questions are relatively standard. However, without further assumptions on the form of the reaction terms, the analysis of propagation phenomena for systems is highly challenging. It is also very interesting from both a mathematical and modeling perspective since more involved and realistic behaviors are expected to happen in systems. Our main objective is precisely to develop techniques for systems that lack such comparison structures in order to provide insights on how interactions among species (including competition and prey-predator relationships) influence each other's spreading dynamics, and may even lead to the possibility of diffusion driven instabilities. Below, we have identified four outstanding problems which constitute the main directions of the Indyana project: - analysis of stage-invasion fronts and locking mechanisms; - analysis of spreading dynamics in heterogeneous and cross-diffusion systems; - analysis of spiraling dynamics in reaction-diffusion systems; - threshold of propagation in reaction-diffusion systems with applications. These are well identified problems in the reaction-diffusion community and present a very active field of research at which our group is at the leading edge. These projects are largely interconnected.