  This thesis pretends to be another step in the development of numerical
research in disordered systems. Specifically, we will focus on spin glasses
which have demonstrated to be a fertile field from both, experimental and
theoretical approaches. Throughout this thesis, we will discuss a variety of
interesting phenomenons and we will also open new avenues to previously
unexplored effects in the context of spin glasses. However, without a doubt,
the leitmotiv conducting this thesis is the role of numerical simulations as a
valuable tool to explore spin-glass physics.
  This thesis is organized into five different parts. The first part,
containing the Chapter 1, is focused on introducing the spin glasses to the
reader.
  The second part, containing the Chapter 2, is dedicated to discussing the
metastate.
  The third part, shaped by the Chapter 3 and Chapter 4, is devoted to studying
the off-equilibrium dynamics in spin glasses. Specifically, Chapter 3 is
focused on discussing the growth of the coherence length in spin glasses, a key
quantity that characterizes the off-equilibrium evolution of those systems. In
Chapter 4 we will discuss an interesting phenomenon: the Mpemba effect.
  The fourth part, containing the Chapter 5, Chapter 6 and Chapter 7 is devoted
to study the Temperature Chaos phenomenon in spin glasses. In Chapter 5 we
introduce the main historical results on Temperature Chaos, from its origins to
the last steps. In Chapter 6 we study equilibrated spin glasses and we
characterize the Temperature Chaos from a static and a dynamical point of view.
In Chapter 7 we tackle the problem of characterizing Temperature Chaos in
off-equilibrium dynamics.
  The fifth and last part of the main body of the thesis corresponds to the
conclusions.
