  We investigate the spectral properties of the product of $M$ complex
non-Hermitian random matrices that are obtained by removing $L$ rows and
columns of larger unitary random matrices uniformly distributed on the group
${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or random
contractions. We first derive the joint probability distribution for the
eigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by a
standard determinantal point process in the complex plane. The weight however
is non-standard and can be expressed in terms of the Meijer G-function. The
explicit knowledge of all eigenvalue correlation functions and the
corresponding kernel allows us to take various large $N$ (and $L$) limits at
fixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condense
on a domain inside the unit circle. At the edge and in the bulk we find the
same universal microscopic kernel as for a single complex non-Hermitian matrix
from the Ginibre ensemble. At the origin we find the same new universality
classes labelled by $M$ as for the product of $M$ matrices from the Ginibre
ensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinity
leads to weak non-unitarity, with most eigenvalues on the unit circle as for
unitary matrices. Here we find a new microscopic edge kernel that generalizes
the known results for M=1. We briefly comment on the case when each product
matrix results from a truncation of different size $L_j$.
