  Elicitability is a property of $\mathbb{R}^k$-valued functionals defined on a
set of distribution functions. These functionals represent statistical
properties of a distribution, for instance its mean, variance, or median. They
are called elicitable if there exists a scoring function such that the expected
score under a distribution takes its unique minimum at the functional value of
this distribution. If such a scoring function exists, it is called strictly
consistent for the functional. Motivated by the recent findings of Fissler and
Ziegel concerning higher order elicitability, this thesis reviews the most
important results, examples, and applications which are found in the relevant
literature. Moreover, we also contribute our own examples and findings in order
to give the reader a well-founded overview of the topic as well as of the most
used tools and techniques. We include necessary and sufficient conditions for
strictly consistent scoring functions, several elicitable as well as
non-elicitable functionals and the use of elicitability in forecast comparison,
regression, and estimation. Special emphasis is placed on quantitative risk
management and the result that Value at Risk and Expected Shortfall are jointly
elicitable.
