  The first mass-estimate of an exoplanet around a Sun-like star, 51 Peg b and
the first radius measurement of an exoplanet, HD209458b pointed to the
challenges of understanding the atmosphere, interior, and evolution of
exoplanets including the possibility of mass loss of planets on close-orbits
that are exposed to strong irradiation. These discoveries raised the question
of heating and inflation mechanisms, and of the nature of these objects in
terms of composition compared to the known planets in the Solar system. The
field of exoplanet interior modeling was born. Here, we outline and discuss
current big science questions: (i) What is the amount of heavy elements in a
planet and do all planets possess an iron-rock core? We suggest that a
promising and novel approach for exoplanets can be measuring their tidal
response in form of the Love numbers h2 and k2. (ii) How much and through what
mechanisms are the interiors of planets heated or delayed from cooling? Many
strongly irradiated gaseous planets require an additional heat source to
explain their large radii. (iii) What is the origin of the observed populations
in the radius-period diagram? Objects in and along the radius valley are
excellent targets to study planetary formation and evaporation. (iv) What does
the composition of rocky planets tell us about their formation? Planets more
iron-rich than Mercury are found, as well as planets that if rocky, are
depleted in iron with respect to Earth. We do not have yet a reliable formation
theory that would explain their existence.
