In the first part of the present thesis, we derive a theory for quantifying entanglement of mixed bipartite quantum states. We derive upper and lower bounds for the concurrence of quantum states in arbitrary finite dimensions, such as to confine its actual value to a finite interval. We test these estimates for various sets of states with very satisfactory results. In particular, our lower bound detects entangled states with positive partial transpose. In view of the specific requirements of laboratory experiments, we derive an approximate expression for the concurrence of almost pure - i.e. weakly mixed - states. Comparison of this quasi-pure approximation with the above upper and lower bounds shows that its range of validity even comprises states with relatively large mixing. Finally, we propose a generalised concurrence for multipartite mixed states. For its quantitative characterisation, we can use the same strategies as in our treatment of bipartite systems. These novel techniques for a quantitative description of the entanglement of mixed states allow to monitor the production and the decay of entanglement under coherent and incoherent forcing - as we show by applying our theory to a realistic scenario of ion trap experiments. In the second part of the thesis, we introduce suitably defined quasi probability representations such as to quantify the entanglement of pure bipartite states, with an immediate generalisation for pure states of multipartite systems. It is shown that the statistical moments as well as various entropies of these representations - characterising their localisation properties - are non-increasing under local operations and classical communication, hence that they are proper entanglement monotones.