  We establish, under a moment matching hypothesis, the local universality of
the correlation functions associated with products of $M$ independent iid
random matrices, as $M$ is fixed, and the sizes of the matrices tend to
infinity. This generalizes an earlier result of Tao and the third author for
the case $M=1$.
  We also prove Gaussian limits for the centered linear spectral statistics of
products of $M$ independent iid random matrices. This is done in two steps.
First, we establish the result for product random matrices with Gaussian
entries, and then extend to the general case of non-Gaussian entries by another
moment matching argument. Prior to our result, Gaussian limits were known only
for the case $M=1$. In a similar fashion, we establish Gaussian limits for the
centered linear spectral statistics of products of independent truncated random
unitary matrices. In both cases, we are able to obtain explicit expressions for
the limiting variances.
  The main difficulty in our study is that the entries of the product matrix
are no longer independent. Our key technical lemma is a lower bound on the
least singular value of the translated linearization matrix associated with the
product of $M$ normalized independent random matrices with independent and
identically distributed subgaussian entries. This lemma is of independent
interest.
