  This thesis presents the Conditional Value-at-Risk concept and combines an
analysis that covers its application as a risk measure and as a vector norm.
For both areas of application the theory is revised in detail and examples are
given to show how to apply the concept in practice.
  In the first part, CVaR as a risk measure is introduced and the analysis
covers the mathematical definition of CVaR and different methods to calculate
it. Then, CVaR optimization is analysed in the context of portfolio selection
and how to apply CVaR optimization for hedging a portfolio consisting of
options. The original contributions in this part are an alternative proof of
Acerbi's Integral Formula in the continuous case and an explicit programme
formulation for portfolio hedging.
  The second part first analyses the Scaled and Non-Scaled CVaR norm as new
family of norms in $\mathbb{R}^n$ and compares this new norm family to the more
widely known $L_p$ norms. Then, model (or signal) recovery problems are
discussed and it is described how appropriate norms can be used to recover a
signal with less observations than the dimension of the signal. The last
chapter of this dissertation then shows how the Non-Scaled CVaR norm can be
used in this model recovery context. The original contributions in this part
are an alternative proof of the equivalence of two different characterizations
of the Scaled CVaR norm, a new proposition that the Scaled CVaR norm is
piecewise convex, and the entire \autoref{chapter:Recovery_using_CVaR}. Since
the CVaR norm is a rather novel concept, its applications in a model recovery
context have not been researched yet. Therefore, the final chapter of this
thesis might lay the basis for further research in this area.
