In this short review, we summarize our present understanding (and non-understanding) of exoplanet formation, structure and evolution, in the light of the most recent discoveries. Recent observations of transiting massive brown dwarfs seem to remarkably confirm the predicted theoretical mass-radius relationship in this domain. This mass-radius relationship provides, in some cases, a powerful diagnostic to distinguish planets from brown dwarfs of same mass, as for instance for Hat-P-20b. If confirmed, this latter observation shows that planet formation takes place up to at least 8 Jupiter masses. Conversely, observations of brown dwarfs down to a few Jupiter masses in young, low-extinction clusters strongly suggests an overlapping mass domain between (massive) planets and (low-mass) brown dwarfs, i.e. no mass edge between these two distinct (in terms of formation mechanism) populations. At last, the large fraction of heavy material inferred for many of the transiting planets confirms the core-accretion scenario as been the dominant one for planet formation.

Thanks to the combination of transit photometry and radial velocity doppler measurements, we are now able to constrain theoretical models of the structure and evolution of objects in the whole mass range between icy giants and stars, including the giant planet/brown dwarf overlapping mass regime (Leconte et al. 2009). In the giant planet mass range, the significant fraction of planets showing a larger radius than predicted by the models suggests that a missing physical mechanism which is either injecting energy in the deep convective zone or reducing the net outward thermal flux is taking place in these objects. Several possibilities have been suggested for such a mechanism:

• • downward transport of kinetic energy originating from strong winds generated at the planet's surface (Showman & Guillot 2002),

• • enhanced opacity sources in hot-Jupiter atmospheres (Burrows et al. 2007),

• • ohmic dissipation in the ionized atmosphere (Batygin & Stevenson 2010),

• • (inefficient) layered or oscillatory convection in the planet's interior (Chabrier & Baraffe 2007),

• • Tidal heating due to circularization of the orbit, as originally suggested by Bodenheimer, Lin & Mardling (2001).

We review the various theories which have been proposed along the years to explain the origin of the stellar initial mass function. We pay particular attention to four models, namely the competitive accretion and the theories based respectively on stopped accretion, MHD shocks and turbulent dispersion. In each case, we derive the main assumptions and calculations that support each theory and stress their respective successes and failures or difficulties.

Radiative feedback and magnetic field are understood to have a strong impact on the protostellar collapse. We present high resolution numerical calculations of the collapse of a 1 M⊙ dense core in solid body rotation, including both radiative transfer and magnetic field. Using typical parameters for low-mass cores, we study thoroughly the effect of radiative transfer and magnetic field on the first core formation and fragmentation. We show that including the two aforementioned physical processes does not correspond to the simple picture of adding them separately. The interplay between the two is extremely strong, via the magnetic braking and the radiation from the accretion shock.

We briefly review the crucial effect that convection plays in the evolution, interior and atmospheric physics of brown dwarfs. In particular, we discuss: (1) the role of convection in the interior structure and evolution of brown dwarfs (BDs); (2) the importance of convection in the atmospheres of ultra-cool objects, at both high and low gravities (i.e., in both young and old BDs); and (3) the role of convection/turbulence in the formation and settling of dust grains, and non-equilibrium chemistry, in BD atmospheres.

We analyse evolutionary tracks at young ages for low mass stars with masses m ≤ 1.4 M⊙ and brown dwarfs down to one mass of Jupiter. We analyse current theoretical uncertainties due to initial conditions. Simple tests on initial conditions show the high uncertainties of models at ages ≲ 1 Myr.

We present evolutionary models for young low mass stars (m ≤ 1M⊙) based on recent improvement of the theory: equation of state, atmosphere models, etc. We concentrate on early evolutionary phases from the initial deuterium burning phase to the zero-age Main Sequence. Evolutionary models for young brown dwarfs are also presented. We discuss the uncertainties of the present models. We analyse the difficulties arising when comparing models with observations for very young objects, in particular concerning the problem of reddening.

We study equilibrium properties of partially ionized hydrogen atmospheres and subphotospheric layers of weakly (with magnetic field B ≪ 109G) and strongly (B ≫ 1010G) magnetized neutron stars. In both weak- and strong-field cases, the ionization degree, atomic occupation numbers, and equation of state are calculated. These results are used to calculate opacities of neutron-star atmospheres.

Accurate modeling of the mechanical and thermal properties of very-lowmass stars (VLMS), Brown Dwarfs (BD) and Extra-solar Giant Planets (EGP) is of prior importance for a wide range of physical and astrophysical problems, from the fundamental physics point of view to the astrophysical and cosmological implications. They provide natural laboratories to test the different theories, equations of state, nuclear reaction rates, model atmospheres aimed at describing the physics of dense and cool objects. They represent the largest stellar population in the Galaxy, and thus provide a substantial contribution to the Galactic (disk) mass budget. Finally they represent one of the most intriguing questions in our understanding of the formation of star-like objects: are planet and star formation processes really different? Is there, and if so what is, a minimum mass for the formation of star-like objects? This field has blossomed recently with the discovery of several brown dwarfs (Nakajima et al, 1995; Rebolo et al., 1995) and numerous exoplanets since 51 Pegasi (Mayor and Queloz 1995; Mayor, this conference), which provide important information to challenge the theory.