In recent years a large number of high-resolution ALMA images of protoplanetary disks were published. It is striking how cleanly structured these disks appear to be: they feature nearly perfect rings of dust, with every now and then an elliptic blob, and sometimes m=2 “grand design spirals” (among several other interesting features). What do these structures tell us about the physics of protoplanetary disks and planet formation? In this talk I will discuss several attempts, some successful, others not, at explaining these features and finding out what causes them. This involves models of varying complexity, from simple semi-analytic models to full-scale multidimensional gas dynamic models.