We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPT coefficients in 4D conformal and N = 1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N = 1 superconformal theories, we place strong bounds on dim(Phi(dagger)Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi(dagger)Phi) <= 2 dim(Phi) near dim(Phi) similar or equal to 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N = 1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.