During the first million years of evolution, nascent planetary systems are embedded in dense disk-shaped clouds of gas. These "circumstellar disks" are home to a myriad of hydrodynamical processes, which bring about turbulence and the emergence of viscous-like behavior, enabling accretion of gas onto the forming star. Meanwhile, micron-sized dust grains embedded in the disk are growing into pebbles and rocks. Turbulence has a positive effect on these small solids, concentrating them into transient high pressure regions for long enough to achieve gravitational collapse through pebble accretion into km-sized bodies, forming the first planetesimals. Giant storm systems in the disk, similar to Jupiter's Great Red Spot, may exist in quiescent zones of the disk. These are even more prone to collecting solid material, producing the first terrestrial planets and cores of giant planets. In this talk I will discuss the state of the art and recent advances in the field of planet formation, as well as pressing problems such as the structure observed in high resolution images of circumstellar disks, and how to interpret them.