(the *supergravity landscape*).

In this talk, which is based on Refs. [@Kallosh:1993wx; @Bellorin:2005hy; @Bellorin:2005zc], we are going to review some recent general results on the classification of supersymmetric String Theory vacua and new techniques that can be used to find them, presenting some particular results on the classification of the supersymmetric vacua of the toroidally compactified Heterotic String Theory ($N=4,d=4$ SUGRA). First, we are going to define what is a supersymmetric configuration and its symmetry superalgebra, describing some useful special identities that they satisfy (*Killing spinor identities*). Then we will move on to define the problem of finding all the supersymmetric configurations of a given supergravity theory *Tod’s problem* and we will explain the strategy to solve it in most (4-dimensional) cases. Finally, we will consider the case of $N=4,d=4$ supergravity.

Supersymmetric configurations and solutions
===========================================

Supersymmetric configurations[^2] (a.k.a. configurations with residual or unbroken or preserved supersymmetry) are classical bosonic configurations of supergravity (SUGRA) theories which are invariant under some supersymmetry transformations. Let us see what this definition implies.

Generically, the supersymmetry transformations take, schematically, the form

$$\label{eq:genericsusyrules}
  \delta_{\epsilon} \phi^{b}
 \sim \bar{\epsilon} \phi^{f}\, ,
\hspace{1cm}
  \delta_{\epsilon} \phi^{f}
 \sim \partial \epsilon
+\phi^{b}\epsilon\, ,$$

where $\phi^{b}$ stands for bosonic fields (or products 