  We discuss the product of $M$ rectangular random matrices with independent
Gaussian entries, which have several applications including wireless
telecommunication and econophysics. For complex matrices an explicit expression
for the joint probability density function is obtained using the
Harish-Chandra--Itzykson--Zuber integration formula. Explicit expressions for
all correlation functions and moments for finite matrix sizes are obtained
using a two-matrix model and the method of bi-orthogonal polynomials. This
generalises the classical result for the so-called Wishart--Laguerre Gaussian
unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for
the product of square matrices. The correlation functions are given by a
determinantal point process, where the kernel can be expressed in terms of
Meijer $G$-functions. We compare the results with numerical simulations and
known results for the macroscopic level density in the limit of large matrices.
The location of the endpoints of support for the latter are analysed in detail
for general $M$. Finally, we consider the so-called ergodic mutual information,
which gives an upper bound for the spectral efficiency of a MIMO communication
channel with multi-fold scattering.
