  We discuss how a spin system, which is subject to quenched disorder, might
exhibit multicritical behaviors at criticality if the distribution of the
impurities is arbitrary. In order to provide realistic candidates for such
multicritical behaviors, we discuss several generalizations of the standard
randomly diluted Ising's universality class adopting the $\epsilon$-expansion
close to several upper critical dimensions. In the presentation, we spend a
special effort in bridging between CFT and RG results and discuss in detail the
computation of quantities, which are of prominent interest in the case of
logarithmic CFT.
