  Recently, sparsity has become a key concept in various areas of applied
mathematics, computer science, and electrical engineering. One application of
this novel methodology is the separation of data, which is composed of two (or
more) morphologically distinct constituents. The key idea is to carefully
select representation systems each providing sparse approximations of one of
the components. Then the sparsest coefficient vector representing the data
within the composed - and therefore highly redundant - representation system is
computed by $\ell_1$ minimization or thresholding. This automatically enforces
separation. This paper shall serve as an introduction to and a survey about
this exciting area of research as well as a reference for the state-of-the-art
of this research field. It will appear as a chapter in a book on "Compressed
Sensing: Theory and Applications" edited by Yonina Eldar and Gitta Kutyniok.
