  LDA+DMFT is a novel computational technique for ab initio investigations of
real materials with strongly correlated electrons, such as transition metals
and their oxides. It combines the strength of conventional band structure
theory in the local density approximation (LDA) with a modern many-body
approach, the dynamical mean-field theory (DMFT). In the last few years
LDA+DMFT has proved to be a powerful tool for the realistic modeling of
strongly correlated electronic systems. In this paper the basic ideas and the
set-up of the LDA+DMFT(X) approach, where X is the method used to solve the
DMFT equations, are discussed. Results obtained with X=QMC (quantum Monte
Carlo) and X=NCA (non-crossing approximation) are presented and compared. By
means of the model system La$_{1-x}$Sr$_{x}$TiO$_{3}$ we show that the method X
matters qualitatively and quantitatively. Furthermore, we discuss recent
results on the Mott-Hubbard metal-insulator transition in the transition metal
oxide V$_2$O$_3$ and the $\alpha$-$\gamma$ transition in the 4f-electron system
Ce.
