  In this paper we reconsider the problem of the Euler parametrization for the
unitary groups. After constructing the generic group element in terms of
generalized angles, we compute the invariant measure on SU(N) and then we
determine the full range of the parameters, using both topological and
geometrical methods. In particular, we show that the given parametrization
realizes the group $SU(N+1)$ as a fibration of U(N) over the complex projective
space $\mathbb{CP}^n$. This justifies the interpretation of the parameters as
generalized Euler angles.
