  Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of
its universal cover. For example, if $G$ is $k$-regular then
$\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering
(a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from
above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan
$r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus,
Spielman and Srivastava (2013).
  Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by
elements of the symmetric group $S_{r}$. We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
  In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
  Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the $r$-th matching polynomial of $G$
to be the average matching polynomial of all $r$-coverings of $G$. We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$.
