Adrienn Ruzsinszky Temple University Kohn-Sham density functional theory (KS-DFT) delivers the ground-state energy and electron density of a many-electron system from the selfconsistent solution of one-electron Schroedinger equations. Only the density functional for the exchange-correlation energy needs to be approximated. On the ladder of approximations, the first three rungs (local spin density approximation, generalized gradient approximation or GGA, and meta-GGA) are semilocal, constructing this energy density from the local electron density, its gradient, and the positive orbital kinetic energy density. The fourth rung is truly nonlocal, employing also the exact exchange energy density. Semilocal density functionals like the local spin density and generalized gradient approximations are known to fail to predict the correct dissociation limit for radical molecules, reaction energy barrier heights and response coefficients of long-chain molecules. This failure of the semilocal functionals arises from the self-interaction error present in approximate density functionals. The Perdew-Zunger self-interaction correction (PZ-SIC), its scaled-down versions, global and local hybrids achieve significant improvement in these cases. As an alternative to these approaches, the random phase approximation (RPA) on the fifth rung has recently gained interest as a possible tool to treat challenges for semilocal density functionals. RPA is self-interaction free in its exchange part, but suffers from serious self-correlation. In this lecture I will show all former and recent improvements from full or scaled-down self-interaction corrections, the concepts of one – and many-electron self-interaction freedom, global and local hybrids, RPA and its corrections. I will discuss the possibilities and limitations of all these approaches.