  Quantum entanglement is at the heart of many tasks in quantum information.
Apart from simple cases (low dimensions, few particles, pure states), however,
the mathematical structure of entanglement is not yet fully understood. This
tutorial is an introduction to our present knowledge about how to decide
whether a given state is separable or entangled, how to characterize
entanglement via witness operators, how to classify entangled states according
to their usefulness (i.e. distillability), and how to quantify entanglement
with appropriate measures.
