  Impurities, defects, and other types of imperfections are ubiquitous in
realistic quantum many-body systems and essentially unavoidable in solid state
materials. Often, such random disorder is viewed purely negatively as it is
believed to prevent interesting new quantum states of matter from forming and
to smear out sharp features associated with the phase transitions between them.
However, disorder is also responsible for a variety of interesting novel
phenomena that do not have clean counterparts. These include Anderson
localization of single particle wave functions, many-body localization in
isolated many-body systems, exotic quantum critical points, and "glassy" ground
state phases. This brief review focuses on two separate but related subtopics
in this field. First, we review under what conditions different types of
randomness affect the stability of symmetry-broken low-temperature phases in
quantum many-body systems and the stability of the corresponding phase
transitions. Second, we discuss the fate of quantum phase transitions that are
destabilized by disorder as well as the unconventional quantum Griffiths phases
that emerge in their vicinity.
