  We present several arguments which favor the scenario of two coexisting
families of compact stars: hadronic stars and quark stars. Besides the well
known hyperon puzzle of the physics of compact stars, a similar puzzle exists
also when considering delta resonances. We show that these particles appear at
densities close to twice saturation density and must be therefore included in
the calculations of the hadronic equation of state. Such an early appearance is
strictly related to the value of the L parameter of the symmetry energy that
has been found, in recent phenomenological studies, to lie in the range
$40<L<62$ MeV. We discuss also the threshold for the formation of deltas and
hyperons for hot and lepton rich hadronic matter. Similarly to the case of
hyperons, also delta resonances cause a softening of the equation of state
which makes it difficult to obtain massive hadronic stars. Quark stars, on the
other hand, can reach masses up to $2.75 M_{\odot}$ as predicted by
perturbative QCD calculations. We then discuss the observational constraints on
the masses and the radii of compact stars. The tension between the precise
measurements of high masses and the indications of the existence of very
compact stellar objects (with radii of the order of $10$ km) is relieved when
assuming that very massive compact stars are quark stars and very compact stars
are hadronic stars. Finally, we discuss recent interesting measurements of the
eccentricities of the orbits of millisecond pulsars in low mass X-ray binaries.
The high values of the eccentricities found in some cases could be explained by
assuming that the hadronic star, initially present in the binary system,
converts to a quark star due to the increase of its central density.
