  This thesis develops recent work on the so called Volume-Complexity and
Action-Complexity conjectures. According to this family of proposals, geometric
quantities can be defined in some holographic gravitational theories that can
be mapped with the concept of quantum complexity for states in a dual
quantum-mechanical theory.
  In this work, we review the original motivations for the use of
quantum-information theory in the search of a theory of quantum gravity, and
argue in favour of holographic complexity as a promising new tool that could
play a key role in the elucidation of the properties of black holes.
  After this introduction, we devote some time to the study of `exotic'
thermodynamical systems of diverse origin, confronting the conjectures with
expectations and seeking for new behaviours of holographic complexity that
could help us understand or refine the existing proposals.
  Next, we turn our attention to the study of holographic complexity for
singular spacetimes, defining slightly modified versions of the conjecture that
are well adapted to singularities and searching for universal behaviours of
complexity dynamics within these setups.
  Finally, we finish with some speculations about the relation between
holographic complexity and older characterization criteria for singularities in
general relativity.
