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We discuss how a spin system, which is subject to quenched disorder, mightexhibit multicritical behaviors at criticality if the distribution of theimpurities is arbitrary. In order to provide realistic candidates for suchmulticritical behaviors, we discuss several generalizations of the standardrandomly diluted Ising's universality class adopting the $\epsilon$-expansionclose to several upper critical dimensions. In the presentation, we spend aspecial effort in bridging between CFT and RG results and discuss in detail thecomputation of quantities, which are of prominent interest in the case oflogarithmic CFT.

RG and logarithmic CFT multicritical properties of randomly diluted Ising models