The first direct detection of gravitational waves has opened a new window to study the Universe and would probably start a new era: the gravitational wave Astronomy. Gravitational waves emitted by compact objects like neutron stars could provide significant information about their structure, composition and evolution.In this paper we calculate, using the relativistic Cowling approximation, the oscillations of compact stars focusing on hybrid stars, with and without a mixed phase in their cores. We study the existence of a possible hadron-quark phase transition in the central regions of neutron stars and the changes it produces on the gravitational modes frequencies emitted by these stars. We pay particular attention to the g-modes, which are extremely important as they could signal the existence of pure quark matter inside neutron stars. Our results show a relationship between the frequency of the g-modes and the constant speed of sound parametrization for the quark matter phase. We also show that the inclusion of color superconductivity produces an increase on the oscillation frequencies.We propose that observations of g-modes with frequencies f g between 1 kHz and 1.5 kHz should be interpreted as an evidence of a sharp hadron-quark phase transition in the core of a compact object.

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We study the influence of strong magnetic fields in hybrid stars, composed by hadrons and a pure quark matter core, and analyse their structure and stability as well as some possible evolution channels due to the magnetic field decay. Using an ad-hoc parametrisation of the magnetic field strength and taking into account Landau-quantization effects in matter, we calculate hybrid magnetised equations of state and some associated quantities, such as particle abundances and matter magnetisation, for different sets of parameters and different magnetic field strengths. Moreover, we compute the magnetised stable stellar configurations, the mass versus radius and the gravitational mass versus central energy density relationships, the gravitational mass versus baryon mass diagram, and the tidal deformability. Our results are in agreement with both, the ∼ 2M pulsars and the data obtained from GW170817. In addition, we study the stability of stellar configurations assuming that slow and rapid phase transitions occur at the sharp hadron-quark interface. We find that, unlike in the rapid transition scenario, where ∂ M/∂ c < 0 is a sufficient condition for instability, in the slow transition scenario there exists a connected extended stable branch beyond the maximum mass star, for which ∂ M/∂ c < 0. Finally, analysing the gravitational mass versus baryon mass relationship, we have calculated the energy released in transitions between stable stellar configurations. We find that the inclusion of the magnetic field and the existence of new stable branches allows the possibility of new channels of transitions that fulfil the energy requirements to explain Gamma Ray Bursts. From a model-theoretical point of view, magnetars are magnetised NSs (M ∼ 2 M , R ∼ 15 km (Glendenning 1997)) with an external solid crust and an internal extremely dense