  The numerical generation of random quantum states (RQS) is an important
procedure for investigations in quantum information science. Here we review
some methods that may be used for performing that task. We start by presenting
a simple procedure for generating random state vectors, for which the main tool
is the random sampling of unbiased discrete probability distributions (DPD).
Afterwards the creation of random density matrices is addressed. In this
context we first present the standard method, which consists in using the
spectral decomposition of a quantum state for getting RQS from random DPDs and
random unitary matrices. In the sequence the Bloch vector parametrization
method is described. This approach, despite being useful in several instances,
is not in general convenient for RQS generation. In the last part of the
article we regard the overparametrized method (OPM) and the related Ginibre and
Bures techniques. The OPM can be used to create random positive semidefinite
matrices with unit trace from randomly produced general complex matrices in a
simple way that is friendly for numerical implementations. We consider a
physically relevant issue related to the possible domains that may be used for
the real and imaginary parts of the elements of such general complex matrices.
Subsequently a too fast concentration of measure in the quantum state space
that appears in this parametrization is noticed.
