**Abstract: **Compact objects in general and neutron stars (NSs) in particular open a window to some of the most extreme physics we can find in nature. On the one hand in the interior of NSs we can find matter in very extreme densities, exceeding nuclear densities and anything we can probe in the laboratory, while on the other hand NSs are related to the strongest gravitational fields next only to those found in black holes. Therefore studying NSs gives us access to both supranuclear densities and strong gravity and can be used to get information and test our theories of matter (equation of state) and gravity. The properties of the structure of NSs are encoded on the spacetime around them and by studying the astrophysical processes that take place around NSs we can map that spacetime.Here we will talk about the properties of NS structure and how they are related to the properties of the spacetime around them. We will also talk about the relation of these properties to astrophysical observables. Finally we will extend our discussion to the case of scalar-tensor theory of gravity.

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This seminar is supported by Portuguese Funds through the CIDMA - Centre for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology ("FCT" - Fundação para a Ciência e a Tecnologia), within the project UID/MAT/04106/2013.