  We consider Euclidean Conformal Field Theories perturbed by quenched
disorder, namely by random fluctuations in their couplings. Such theories are
relevant for second-order phase transitions in the presence of impurities or
other forms of disorder. Theories with quenched disorder often flow to new
fixed points of the renormalization group. We begin with disorder in free field
theories. Imry and Ma showed that disordered free fields can only exist for
d>4. For d>4 we show that disorder leads to new fixed points which are not
scale-invariant. We then move on to large-N theories (vector models or gauge
theories in the `t Hooft limit). We compute exactly the beta function for the
disorder, and the correlation functions of the disordered theory. We generalize
the results of Imry and Ma by showing that such disordered theories exist only
when disorder couples to operators of dimension \Delta > d/4. Sometimes the
disordered fixed points are not scale-invariant, and in other cases they have
unconventional dependence on the disorder, including non-trivial effects due to
irrelevant operators. Holography maps disorder in conformal theories to
stochastic differential equations in a higher-dimensional space. We use this
dictionary to reproduce our field theory results. We also study the leading 1/N
corrections, both by field theory methods and by holography. These corrections
are particularly important when disorder scales with the number of degrees of
freedom.
