The ground-state energy and properties of any many-electron atom ormolecule may be rigorously computed by variationally computing thetwo-electron reduced density matrix rather than the many-electronwavefunction. While early attempts fifty years ago to compute theground-state 2-RDM directly were stymied because the 2-RDM must beconstrained to represent an N-electron wavefunction, recentadvances in theory and optimization have made direct computation ofthe 2-RDM possible. The constraints in the variational calculationof the 2-RDM require a special optimization known as a semidefiniteprogramming. Development of first-order semidefinite programmingfor the 2-RDM method has reduced the computational costs of thecalculation by orders of magnitude [Mazziotti, Phys. Rev. Lett.93 (2004) 213001]. The variational 2-RDM approach is effective atcapturing multi-reference correlation effects that are especiallyimportant at non-equilibrium molecular geometries. Recent work on2-RDM methods will be reviewed and illustrated with particularemphasis on the importance of advances in large-scale semidefiniteprogramming.