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The last years have brought upon us a golden age of observationalgravitational physics. The several observations by the LIGO/Virgo/KAGRAcollaboration about gravitational waves and by the EHT collaboration about theshadow and lensing of light around the supermassive black hole in the centre ofM87 will point the scientific community in the correct direction to find ananswer to the Kerr hypothesis. In order to follow that direction, thesystematic construction and analysis of the physical properties of solutionswithin General Relativity with additional fields or within modified theories ofgravity is necessary. In this thesis, we shall provide such construction andanalysis for compact objects within (complex-)Einstein-Klein-Gordon theory withvarious scalar potentials and within a particular scalar-tensor theory -- theshift-symmetric Horndesky theory. After a brief introduction to some key topicsthat shall be useful throughout this thesis, we present a discussion about thehorizon geometry of Kerr black holes with and without scalar hair. We follow upwith the construction and study of the same hairy solutions discussed in theprevious chapter but with higher azimuthal harmonic indexes. In the followingtwo chapters, we introduce a different scalar potential based on the QuantumChromodynamic axion potential and obtain and study both horizonless compactobjects and black holes. We then go to the shift-symmetric Horndeski theory,where we perform similar constructions and analyses to the ones alreadymentioned. Lastly, we derive a relation between the radial stability oflight-rings and timelike circular orbits around them. We follow up with a studyon how efficient it is the conversion of gravitational energy in radiation as atimelike particle falls towards all compact objects studied throughout thisthesis. We end with some conclusions and remarks.

Spinning Black Holes with Scalar Hair and Horizonless Compact Objects within and beyond General Relativity