We discuss the factors influencing the formation and gravitational fragmentation of protostellar discs. We start with a review of how observations of prestellar cores can be analysed statistically to yield plausible initial conditions for simulations of their subsequent collapse. Simulations based on these initial conditions show that, despite the low levels of turbulence in prestellar cores, they deliver primary protostars and associated discs which are routinely subject to stochastic impulsive perturbations; consequently misalignment of the spins and orbits of protostars are common. Also, the simulations produce protostars that collectively have a mass function and binary statistics matching those observed in nearby star-formation regions, but only if a significant fraction of the turbulent energy in the core is solenoidal, and accretion onto the primary protostar is episodic with a duty cycle $\stackrel{>}{\sim }3\,000\,{\rm yr}$. Under this circumstance, a core typically spawns between 4 and 5 protostars, with high efficiency, and the lower mass protostars are mainly formed by disc fragmentation. The requirement that a proto-fragment in a disc lose thermal energy on a dynamical timescale dictates that there is a sweet spot for disc fragmentation at radii $70\,{\rm AU}\stackrel{<}{\sim }R\stackrel{<}{\sim }100\,{\rm AU}$ and temperatures $10\,{\rm K}\stackrel{<}{\sim }T\stackrel{<}{\sim }20\,{\rm K}$, and this might explain the brown dwarf desert.

Disc fragmentation provides an important mechanism for producing low-mass stars in prestellar cores. Here, we describe smoothed particle hydrodynamics simulations which show how populations of prestellar cores evolve into stars. We find the observed masses and multiplicities of stars can be recovered under certain conditions.

First, protostellar feedback from a star must be episodic. The continuous accretion of disc material on to a central protostar results in local temperatures which are too high for disc fragmentation. If, however, the accretion occurs in intense outbursts, separated by a downtime of ~ 104 yr, gravitational instabilities can develop and the disc can fragment.

Second, a significant amount of the cores’ internal kinetic energy should be in solenoidal turbulent modes. Cores with less than a third of their kinetic energy in solenoidal modes have insufficient angular momentum to form fragmenting discs. In the absence of discs, cores can fragment but results in a top-heavy distribution of masses with very few low-mass objects.

Transition discs are protoplanetary discs that show evidence for large holes or wide gaps (with widths comparable to their radii) in their dust component. These discs could be giving us clues about the disc destruction mechanism or hints about the location and time-scales for the formation of planets. However, at the moment there remain key gaps in our theoretical understanding. The vast majority of transition discs are accreting onto their central stars, indicating that—at least close to the star—dust has been depleted from the gas by a very large amount. In this review, we discuss evidence for two distinct populations of transition discs: mm-faint—those with low mm-fluxes, small holes (≲ 10 AU), and low accretion rates (~ 10−10 − 10−9 M⊙ yr−1) and mm-bright—discs with large mm-fluxes, large holes (≳ 20 AU), and high accretion rates ~ 10−8 M⊙ yr−1. MM-faint transition discs are consistent with what would naively be expected from a disc undergoing dispersal; however, mm-bright discs are not, and are likely to be rare and long-lived objects. We discuss the two commonly proposed mechanisms for creating transition discs: photoevaporation and planet–disc interactions, with a particular emphasis on how they would evolve in these models, comparing these predictions to the observed population. More theoretical work on explaining the lack of optically thick, non-accreting transition discs is required in both the photoevaporation and planetary hypothesis, before we can start to use transition discs to constrain models of planet formation. Finally, we suggest that the few discs with primordial looking spectral energy distribution, but serendipitously imaged showing large cavities in the mm (e.g. MWC758 and WSB 60) may represent a hidden population of associated objects. Characterising and understanding how these objects fit into the overall paradigm may allow us to unravel the mystery of transition discs.

The magnetic field plays a central role in the formation and evolution of circumstellar disks. The magnetic field connects the rapidly rotating central region with the outer envelope and extracts angular momentum from the central region during gravitational collapse of the cloud core. This process is known as magnetic braking. Both analytical and multidimensional simulations have shown that disk formation is strongly suppressed by magnetic braking in moderately magnetised cloud cores in the ideal magnetohydrodynamic limit. On the other hand, recent observations have provided growing evidence of a relatively large disk several tens of astronomical units in size existing in some Class 0 young stellar objects. This introduces a serious discrepancy between the theoretical study and observations. Various physical mechanisms have been proposed to solve the problem of catastrophic magnetic braking, such as misalignment between the magnetic field and the rotation axis, turbulence, and non-ideal effect. In this paper, we review the mechanism of magnetic braking, its effect on disk formation and early evolution, and the mechanisms that resolve the magnetic braking problem. In particular, we emphasise the importance of non-ideal effects. The combination of magnetic diffusion and thermal evolution during gravitational collapse provides a robust formation process for the circumstellar disk at the very early phase of protostar formation. The rotation induced by the Hall effect can supply a sufficient amount of angular momentum for typical circumstellar disks around T Tauri stars. By examining the combination of the suggested mechanisms, we conclude that the circumstellar disks commonly form in the very early phase of protostar formation.

It is quite likely that self-gravity will play an important role in the evolution of accretion discs, in particular, those around young stars, and those around supermassive black holes. We summarise, here, our current understanding of the evolution of such discs, focussing more on discs in young stellar system, than on discs in active galactic nuclei. We consider the conditions under which such discs may fragment to form bound objects, and when they might, instead, be expected to settle into a quasi-steady, self-regulated state. We also discuss how this understanding may depend on the mass of the disc relative to the mass of the central object, and how it might depend on the presence of external irradiation. Additionally, we consider whether or not fragmentation might be stochastic, where we might expect it to occur in an actual protostellar disc, and if there is any evidence for fragmentation actually playing a role in the formation of planetary-mass bodies. Although there are still a number of outstanding issue, such as the convergence of simulations of self-gravitating discs, whether or not there is more than one mode of fragmentation, and quite what role self-gravitating discs may play in the planet-formation process, our general understanding of these systems seems quite robust.

Resolved observations are bringing new constraints on the origin of radial gaps in protoplanetary disks. The kinematics, sampled in detail in one case-study, are indicative of non-Keplerian flows, corresponding to warped structures and accretion which may both play a role in the development of cavities. Disk asymmetries seen in the radio continuum are being interpreted in the context of dust segregation via aerodynamic trapping. We summarise recent observational progress, and describe prospects for improvements in the near term.

We study a possibility to detect signatures of brown dwarf companions in a circumstellar disc based on spectral energy distributions. We present the results of spectral energy distribution simulations for a system with a 0.8 M⊙ central object and a companion with a mass of 30 MJ embedded in a typical protoplanetary disc. We use a solution to the one-dimensional radiative transfer equation to calculate the protoplanetary disc flux density and assume, that the companion moves along a circular orbit and clears a gap. The width of the gap is assumed to be the diameter of the brown dwarf Hill sphere. Our modelling shows that the presence of such a gap can initiate an additional minimum in the spectral energy distribution profile of a protoplanetary disc at λ = 10–100 μm. We found that it is possible to detect signatures of the companion when it is located within 10 AU, even when it is as small as 3 MJ. The spectral energy distribution of a protostellar disc with a massive fragment (of relatively cold temperature ~400 K) might have a similar double peaked profile to the spectral energy distribution of a more evolved disc that contains a gap.

The Protoplanetary Discussions conference—held in Edinburgh, UK, from 2016 March 7th–11th—included several open sessions led by participants. This paper reports on the discussions collectively concerned with the multi-physics modelling of protoplanetary discs, including the self-consistent calculation of gas and dust dynamics, radiative transfer, and chemistry. After a short introduction to each of these disciplines in isolation, we identify a series of burning questions and grand challenges associated with their continuing development and integration. We then discuss potential pathways towards solving these challenges, grouped by strategical, technical, and collaborative developments. This paper is not intended to be a review, but rather to motivate and direct future research and collaboration across typically distinct fields based on community-driven input, to encourage further progress in our understanding of circumstellar and protoplanetary discs.

Recent progress in telescope development has brought us different ways to observe protoplanetary disks: interferometers, space missions, adaptive optics, polarimetry, and time- and spectrally-resolved data. While the new facilities have changed the way we can tackle open problems in disk structure and evolution, there is a substantial lack of interconnection between different observing communities. Here, we explore the complementarity of some of the state-of-the-art observing techniques, and how they can be brought together to understand disk dispersal and planet formation.

This paper was born at the ‘Protoplanetary Discussions’ meeting in Edinburgh, 2016. Its goal is to clarify where multi-wavelength observations converge in unveiling disk structure and evolution, and where they challenge our current understanding. We discuss caveats that should be considered when linking results from different observations, or when drawing conclusions from limited datasets (in terms of wavelength or sample). We focus on disk properties that are currently being revolutionized, specifically: the inner disk radius, holes and gaps and their link to large-scale disk structures, the disk mass, and the accretion rate. We discuss how their connections and apparent contradictions can help us to disentangle the disk physics and to learn about disk evolution.