  We shall consider the product of complex random matrices from the independent
complex Ginibre ensembles. The product includes complex matrices $Z_i,
Z_i^\dagger, \, i = 1, \ldots, n$ and $2n$ sources (complex matrices $C_i$ and
$C_i^*$). Any such product can be represented by a chord diagram that encodes
the order of the matrices in the product. We introduce the Euler characteristic
E$^*$ of the chord diagram and show that the spectral correlation functions of
the product generate Hurwitz numbers that enumerate nonequivalent branched
coverings of Riemann surfaces of genus $g^*$. The role of sources is the
generation of branching profiles in critical points which are assigned to the
vertices of the graph drawn on the base surface obtained as a result of gluing
of the $2n$-gon related to the chord diagram in a standard way. Hurwitz numbers
for Klein surfaces may also be obtained by a slight modification of the model.
Namely, we consider $2n+1$ polygon and consider pairing of the extra matrix
$Z_{2n+1}$ with a "Mobius" tau function. Thus, the presented matrix models
labelled by chord diagrams generate Hurwitz numbers for any given Euler
characteristic of the base surface and for any given set of ramification
profiles.
