Substructures in protostellar discs: spirals, gaps and warps

The advent of high resolution imaging of protostellar discs, both in the sub-mm (thanks to ALMA) and in the near infrared, has radically changed our understanding of the evolution of such discs and of the planet formation process occuring within them. While in the past disc were modeled as simplified, axi-symmetric structures, often characterized by simple radial power-law for density and temperature, we now need more advanced modeling, able to describe the substructures observed. Such modeling needs to take into account both the gas component, that dominates the dynamics and the line emission, and the dust, which is responsible for the continuum mm band emission. Here, I present several aspects of such modeling. I will discuss the theory and some hydrodynamical simulations describing: (a) spiral density waves, for example induced by gravitational instabilities in young and massive discs; (b) gaps induced by the presence of a forming planet in the disc, with particular emphasis on the spectacular case of HL Tau, that we have recently successfully modeled; (c) warps, which are expected to develop in circumbinary discs, or in discs where a planet has been put on a very inclined orbit.