Abstract: Classification of superconformal field theories (SCFTs) is an important question in theoretical physics. We would like to (partially) answer this question focusing on SCFTs in four dimensions realized in the fixed points of the renormalization group flow induced by certain deformations of “known” SCFTs. The deformations include all possible relevant deformations and F-term couplings to gauge-singlet chiral multiplets. By “known” we mean we know the superconformal index in enough higher order. With this knowledge our deformation program is well defined and produces various fixed points. As simple examples, we see the deformations of the N=2 Argyres-Douglas theory, and of N=1 supersymmetric SU(2) gauge theory coupled to an adjoint and two fundamental chiral multiplets. In particular, from the latter theory, we find 34 fixed points, which cover all the known (low central charge) SCFTs and the interesting unknown ones, and 36 candidate fixed points possessing unphysical operators, including one with (a, c)=(0.20, 0.22), that need further investigation.