  The last years have brought upon us a golden age of observational
gravitational physics. The several observations by the LIGO/Virgo/KAGRA
collaboration about gravitational waves and by the EHT collaboration about the
shadow and lensing of light around the supermassive black hole in the centre of
M87 will point the scientific community in the correct direction to find an
answer to the Kerr hypothesis. In order to follow that direction, the
systematic construction and analysis of the physical properties of solutions
within General Relativity with additional fields or within modified theories of
gravity is necessary. In this thesis, we shall provide such construction and
analysis for compact objects within (complex-)Einstein-Klein-Gordon theory with
various scalar potentials and within a particular scalar-tensor theory -- the
shift-symmetric Horndesky theory. After a brief introduction to some key topics
that shall be useful throughout this thesis, we present a discussion about the
horizon geometry of Kerr black holes with and without scalar hair. We follow up
with the construction and study of the same hairy solutions discussed in the
previous chapter but with higher azimuthal harmonic indexes. In the following
two chapters, we introduce a different scalar potential based on the Quantum
Chromodynamic axion potential and obtain and study both horizonless compact
objects and black holes. We then go to the shift-symmetric Horndeski theory,
where we perform similar constructions and analyses to the ones already
mentioned. Lastly, we derive a relation between the radial stability of
light-rings and timelike circular orbits around them. We follow up with a study
on how efficient it is the conversion of gravitational energy in radiation as a
timelike particle falls towards all compact objects studied throughout this
thesis. We end with some conclusions and remarks.
