In the last decades, thousands of exoplanets have been detected, revealing a variety of characteristics different from those of the Solar System's planets - the only planets orbiting a main sequence star known until 1995. This enhanced the interest in processes of planet formation and evolution that can help to explain the observed exoplanets' characteristics. This thesis focuses on the formation of planets via core accretion in protoplanetary discs - disc-shaped structures made of dust and gas that form around newborn stars- and in their subsequent evolution as a consequence of disc-planet interaction from a numerical and theoretical perspective.

In the core accretion scenario, planets form in protoplanetary discs by growing the initial μm-sized dust grains up to the size of a planet. The aerodynamic interaction between the dust grains and the disc gas component, however, causes the grains to lose angular momentum and drift inwards; for cm-sized grains, the drift is so fast that they are expected to rapidly go towards the star, becoming unavailable to form planets. This problem is called the `radial drift barrier', and it can potentially be solved by the action of the streaming instability that causes the rapid formation of dust clumps that can later collapse under the action of self-gravity. In this thesis, we investigate the emission of systems undergoing streaming instability that we simulate through 2D local simulations using the hybrid code ATHENA. By comparing the simulated systems before and after particle clumping to the data from the Lupus star forming region in the optically thick fraction - spectral index plane, we find that the action of streaming instability drives the simulations towards the area of the plane occupied by the data. We further analyse the azimuthal brightness asymmetries produced when systems undergoing streaming instability are observed at an inclination angle. We demonstrate that the optically thick fraction exhibits a peak along the minor axis when the disc containing unresolved annular optically thick substructures is inclined and that, for favourable system parameters, these are likely observable by ALMA.

Once a planet is formed, it is subject to mutual gravitational interaction with the host disc, which modifies both the disc structure and the planet's orbital parameters. The second part of this thesis focuses on the migration of massive planets in the planet-dominated regime of Type II migration. By performing long-timescale, live-planet simulations, we revisit previous results about the existence of a direct correlation between the rate of change of the semi-major axis and the torques acting on the planet. We find that such a correlation breaks for live-planet simulations when planet eccentricity is excited, but it is recovered by disentangling the contribution to the torque due to the semi-major axis evolution from that due to the eccentricity evolution. We develop a toy model based on the existence of that correlation. By applying this model to investigate the planet migration in viscously evolving discs, we show that the planets tend to migrate towards a precise location in the disc (`stalling radius'); this effect, combined with the evolution of the disc, causes the planets to distribute in a band around the stalling radius, estimated to be around 1-10 AU, disfavouring the idea of hot Jupiter formation through Type II migration in the planet-dominated regime.