The field of active matter studies materials composed of self-propelled entities, which can be either biological or synthetic. Examples of such materials stretch across length-scales, from nanorobots to humans, and provide a unique plethora of collective behaviors that are inaccessible in equilibrium materials, ranging from highly ordered bird flocks to phase separation in absence of any attraction among particles. The role of theoretical research in active matter is thus not only to explain natural phenomena but also to deliver robust models that allow to unveil which novel phenomenology we should expect. 
 
The talk will start with a brief overview of active matter and of my recent contributions to it. I will then focus on phase separation in active systems. This is one of the most fundamental collective phenomena arising at high density, and it takes place in systems as diverse as bacterial suspensions and biological tissues. I will show that much understanding can be drawn by generalising the classical field theoretical description of liquid-liquid phase separation (Model B, or phi^4 theory) to an active context. After discussing the thermodynamics of such model, I will show that activity induces phase separated phases that have qualitatively new features with respect to those encountered in passive systems. This surprising fact can be rationalised by generalising the concept of interfacial tension, crucial in passive phase separation, to an active context. Next, we will ask how to control these new forms of phase separation in terms of microscopically tunable parameters; although this question is still largely open, I will describe the recent progresses we made by studying minimal models of active particles.