  We consider the singular value statistics of products of independent random
matrices. In particular we compute the corresponding averages of products of
characteristic polynomials. To this aim we apply the projection formula
recently introduced for chiral random matrix ensembles which serves as a short
cut of the supersymmetry method. The projection formula enables us to study the
local statistics where free probability currently fails. To illustrate the
projection formula and underlining the power of our approach we calculate the
hard edge scaling limit of the Meijer G-ensembles comprising the
Wishart-Laguerre (chiral Gaussian), the Jacobi (truncated orthogonal, unitary
or unitray symplectic) and the Cauchy-Lorentz (heavy tail) random matrix
ensembles. All calculations are done for real, complex, and quaternion matrices
in a unifying way. In the case of real and quaternion matrices the results are
completely new and point to the universality of the hard edge scaling limit for
a product of these matrices, too. Moreover we identify the non-linear
$\sigma$-models corresponding to product matrices.
