  It has been shown by Akemann, Ipsen and Kieburg that the squared singular
values of products of $M$ rectangular random matrices with independent complex
Gaussian entries are distributed according to a determinantal point process
with a correlation kernel that admits a representation in terms of Meijer
G-functions. We prove the universality of the local statistics of the squared
singular values, namely, the bulk universality given by the sine kernel and the
edge universality given by the Airy kernel. The proof is based on the
asymptotic analysis for the double contour integral representation of the
correlation kernel. Our strategy can be generalized to deal with other models
of products of random matrices introduced recently and to establish similar
universal results. Two more examples are investigated, one is the product of
$M$ Ginibre matrices and the inverse of $K$ Ginibre matrices studied by
Forrester, and the other one is the product of $M-1$ Ginibre matrices with one
truncated unitary matrix considered by Kuijlaars and Stivigny.
