We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions ∆ of operators in four-dimensional mathcal{N} = 1 superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on ∆. We analyze in detail chiral operators in the left(frac{1}{2}j,0right) Lorentz representation and prove that the ANEC implies the lower bound Delta ge frac{3}{2}j , which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on left(frac{1}{2}j,0right) operators obeying other possible shortening conditions, as well as general left(frac{1}{2}j,0right) operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our mathcal{N} = 1 results for multiplets of mathcal{N} = 2, 4 superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.