  Despite the extreme simplicity in their definition, spin glasses disclose a
wide variety of non-trivial behaviors that are not yet fully understood. In
this thesis we try to shed light on some of them, focusing on one hand on the
search of phase transitions under perturbations of the Hamiltonian, and on the
other hand on the zero-temperature properties and responses to external
stimuli. After introducing spin glasses through a historiographical review, and
useful concepts on phase transitions and numerics, the results of two massive
Monte Carlo campaigns on three-dimensional systems are shown. In the first one
the de Almeida-Thouless transition is studied, and a new finite-size scaling
ansatz is proposed, that accelerates the convergence to the thermodynamic
limit. In the second one the phase diagram of the Heisenberg spin glass with
random exchange anisotropy is reconstructed. In the following part of the
manuscript, surprising features of zero-temperature statics and dynamics of
several spin glass models are found. Among them, correlations between soft
spins that arise spontaneously during avalanches, and the discovery of
localized states that involve the presence of two-level systems.
