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We classify local **N=2** supersymmetric deformations of **4d N=2** superconformal field theories (SCFTs), and discuss how they can deform the moduli space of vacua. We present evidence for a conjecture which governs the behavior of the generic singularities that can occur on the Coulomb branches of **N=2** SCFTs deformed by relevant operators. This conjecture puts strong constraints on the existence of dangerously irrelevant operators and accidental IR flavor symmetries. This leads to a classification of possible Coulomb branch geometries of deformed SCFTs which have planar topology.

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