  The research field of quantum entanglement theory is comparatively new. While
a basic understanding of the most simple systems in question (i.e. bipartite
systems) has been established over the past few decades, multipartite
entanglement still holds many unsolved questions and intriguing riddles. In
particular, it is completely unclear how several concepts from the bipartite
case can be generalised in a meaningful way to multipartite scenarios. In this
work, the main issues of multipartite entanglement detection, characterisation
and classification are discussed. The differences and similarities between the
bipartite and the multipartite situation are reviewed, various possible
generalisations are presented and results are obtained in several areas. The
focus of this work particularly lies on a formalism - the so called HMGH
framework, which has been developed and expanded for the past several years -
which allows for construction of very specific separability criteria, capable
of discriminating between different kinds of multipartite entanglement. By
means of these criteria, the questions of partial separability, genuine
multipartite entanglement and - ultimately - multipartite entanglement
classification (which appear to contain the most striking differences to
bipartite entanglement) are adressed and discussed. In order to illustrate the
theoretical conclusions in these respects, several examples are given from
different (and differently closely related) fields, showing the capabilities,
strengths and weaknesses of the HMGH framework as well as giving insights into
the current status of research in multipartite entanglement theory as a whole.
