Abstract: CR singularities of real 4 -submanifolds in {mathbb C}³ - points where the tangent space is a complex hyperplane - are classified by using holomorphic coordinate changes to transform the quadratic coefficients of the real analytic local defining equations into one of a list of normal forms. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities. The geometry, both locally and globally, is a natural generalization of the well-known case of surfaces in {mathbb C}2 and Bishop's elliptic/hyperbolic classification of CR singular points.