Systems governed by laws of self-organization arise in many different contexts including biology (swarming behavior of animals), social (opinion dynamics, social networks),  technological (cosmology, control, robotics), and others.  A particular challenge in studying such systems is to understand how local communication between “agents” gives rise to emergence of global collective behavior characterized by two basic phenomena — alignment and flocking. In this talk we outline recent developments in the analysis of such emergent phenomena. We will describe several programs of research that has seen the most progress in recent years, with a special focus on the so-called Euler Alignment System, the macroscopic counterpart of the classical Cucker-Smale model. These systems are widely used in technological applications, yet give rise to a special class of fractional parabolic equations which pose a unique set of challenges from the perspective of regularity theory as well as asymptotic behavior.