  Quantum entanglement is one of the core features of quantum theory. While it
is typically revealed by measurements along carefully chosen directions, here
we review different methods based on so-called random or randomized
measurements. Although this approach might seem inefficient at first, sampling
correlations in various random directions is a powerful tool to study
properties which are invariant under local-unitary transformations. Based on
random measurements, entanglement can be detected and characterized without a
shared reference frame between the observers or even if local reference frames
cannot be defined. This overview article discusses different methods using
random measurements to detect genuine multipartite entanglement and to
distinguish SLOCC classes. Furthermore, it reviews how measurement directions
can efficiently be obtained based on spherical designs.
