  In this Ph.D. dissertation we study the emergence of black-hole and wormhole
solutions in the framework of the Einstein-scalar-Gauss-Bonnet (EsGB) theory.
Particularly we study a family of theories where the coupling function
$f(\phi)$ between the scalar field of the theory and the quadratic Gauss-Bonnet
gravitational term has an arbitrary form. At first, we analytically derive that
the aforementioned family of theories may evade the constraints imposed by
Bekenstein's No-Scalar Hair theorems and new solutions for black holes may be
found. Then, using numerical integration methods we find solutions for black
holes for many different forms of the coupling function. Also we derive their
physical characteristics namely their mass, scalar charge, horizon area and
entropy as well. Subsequently, by introducing a cosmological constant in the
theory we investigate the existence of novel black-hole solutions.
Specifically, assuming that the cosmological constant may be positive or
negative we find numerical solutions which are asymptotically de Sitter or
anti-de Sitter. In addition, as in the case of the asymptotically flat black
holes, for each case we derive their physical characteristics. Finally, in the
framework of the EsGB theory we derive novel wormhole solutions. The
Gauss-Bonnet wormholes are traversable, may have a single or a double throat
and do not demand the existence of exotic matter.
