  In agreement with the constantly increasing gravitational wave events, new
aspects of the internal structure of compact stars can be considered. A
scenario in which a first order transition takes place inside these stars is of
particular interest as it can lead, under conditions, to a third
gravitationally stable branch (besides white dwarfs and neutron stars), the
twin stars. The new branch yields stars with the same mass as normal compact
stars but quite different radii. In the present work, we focus on hybrid stars
undergone a hadron to quark phase transition near their core and how this new
stable configuration arises. Emphasis is to be given on the aspects of the
phase transition and its parametrization in two different ways, namely with
Maxwell and Gibbs construction. We systematically study the gravitational mass,
the radius, and the tidal deformability, and we compare them with the
predictions of the recent observation by LIGO/VIRGO collaboration, the GW170817
event, along with the mass and radius limits, suggesting possible robust
constraints. Moreover, we extent the study in order to include rotation effects
on the twin stars configurations. The recent discovery of the fast rotating
supermassive pulsar PSR J0952-0607 triggered the effort to constrain the
equation of state and moreover to examine possible predictions related to the
phase transition in dense nuclear matter. We pay special attention to relate
the PSR J0952-0607 pulsar properties with the twin stars predictions and mainly
to explore the possibility that the existence of such a massive object would
rule out the existence of twin stars. Finally, we discuss the constraints on
the radius and mass of the recently observed compact object within the
supernova remnant HESS J1731-347. The estimations implies that this object is
either the lightest neutron star known, or a star with a more exotic equation
of state.
