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The dream of materials design is to leverage, rather than ignore, our theories of electronic structure and combine them with our increasing computational ability to discover new materials. Beyond its technological implications, the challenge of materials design is also one of great intellectual depth. In principle, we know the fundamental equation needed to model the behavior of a material: it is the Schr\"odinger equation describing electrons moving in the potential of a periodic lattice, mutually interacting via the Coulomb repulsion. Solving this equation is another matter.  In practice, we can classify materials by how well we can solve their corresponding Schr\"odinger equation. For the class of compounds encompassing simple metals, insulators and semiconductors, termed weakly correlated materials, we have a well-developed theory of their excitation spectra called Fermi liquid theory. Additionally, we have From  a practical viewpoint, the  theoretical framework called of  density functional theory (DFT), which (DFT)  naturally lends itself to computational implementations for modelingtheir  properties. Materials which are not well-described by DFT are colloquially termed strongly correlated materials. In order to understand the challenges particular to correlations in materials design, we need to better define what we mean by a correlated material, and we start by examining DFT.  DFT is a workhorse of the materials science community, providing efficient and accurate computations of the total energy and distribution of electrons of a compound, requiring only the coordinates of the atoms in its crystal lattice as input. From the total energy, one can obtain lattice constants, equations of state and the spectrum of lattice vibrations. Furthermore, one can obtain electronic properties such as band gaps, electric polarization and topological numbers, which are by no means trivial for these "simple" compounds. The conceptual idea behind density functional theory DFT  is that for ground state properties, we can use the total electron density $\rho(\vec{r})$ as the fundamental variable in our equations, rather than the complicated quantum many-body wavefunction $\Psi$. Kohn and Sham~\cite{Kohn_1965} recast the theory in a form useful for computation, practical form,  showing that solving for one could compute  the electron exact electronic  density by solving the problem of an auxiliary system of non-interacting electrons  in the presence of a periodic  potential $V_\text{KS}$ $V_\text{KS}(\vec{r})$.  %  It starts with the Kohn-Sham formulation~\cite{Kohn_1965} of density functional theory. It states the existence of a potential $V_{KS}(r)$, which is itself a functional of the density. One should write $V_{KS}(\vec{r})[\{ \rho(\vec{r}') \}]$ to indicate this dependence, but we omit this in the following. The exact (but unknown) functional is such that the solution of the set of self-consistent equations, %  \begin{equation} %  \left[-\nabla^{2}+V_{KS}\left(\vec{r}\right)\right]\psi_{\vec{k}j}\left(\vec{r}\right)=\epsilon_{\vec{k}j}\psi_{\vec{k}j}\left(\vec{r}\right). %  \label{Kohn-Sham} %  \end{equation} %  \begin{equation} %  \sum_{\vec{k}j} |\phi_{\vec{k}j}(\vec{r})|^2f(\epsilon_{\vec{k}j}) = \rho(\vec{r}) %  \label{KS2} %  \end{equation} %  reproduces the density of the solid. It is useful to divide the Kohn-Sham potential into several parts: $ V_{KS} = V_{Hartree}+V_{cryst} +V_{xc}$, where one lumps into $V_{xc}$ exchange and correlation effects beyond Hartree. The eigenvalues $\epsilon_{\vec{k}j} $ of the solution of the self-consistent set of Eq.~\ref{Kohn-Sham} and~\ref{KS2} are not to be interpreted as excitation energies. Instead the excitation spectra should be extracted from the poles of the one particle Green's function: