  The aim of this note is to provide a pedagogical survey of the recent works
by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local
behavior of the eigenvalues of large complex correlated Wishart matrices at the
edges and cusp points of the spectrum: Under quite general conditions, the
eigenvalues fluctuations at a soft edge of the limiting spectrum, at the hard
edge when it is present, or at a cusp point, are respectively described by mean
of the Airy kernel, the Bessel kernel, or the Pearcey kernel. Moreover, the
eigenvalues fluctuations at several soft edges are asymptotically independent.
In particular, the asymptotic fluctuations of the matrix condition number can
be described. Finally, the next order term of the hard edge asymptotics is
provided.
