  We study the roots of generalized Eulerian polynomials via a novel approach.
We interpret Eulerian polynomials as the generating polynomials of a statistic
over inversion sequences. Inversion sequences (also known as Lehmer codes or
subexcedant functions) were recently generalized by Savage and Schuster, to
arbitrary sequences $\s$ of positive integers, which they called $\s$-inversion
sequences.
  Our object of study is the generating polynomial of the {\em ascent}
statistic over the set of $\s$-inversion sequences of length $n$. Since this
ascent statistic over inversion sequences is equidistributed with the descent
statistic over permutations we call this generalized polynomial the
\emph{$\s$-Eulerian polynomial}. The main result of this paper is that, for any
sequence $\s$ of positive integers, the $\s$-Eulerian polynomial has only real
roots.
  This result is first shown to generalize many existing results about the
real-rootedness of various Eulerian polynomials. We then show that it can be
used to settle a conjecture of Brenti, that Eulerian polynomials for all finite
Coxeter groups have only real roots. It is then extended to several
$q$-analogs. We also show that the MacMahon--Carlitz $q$-Eulerian polynomial
has only real roots whenever $q$ is a positive real number confirming a
conjecture of Chow and Gessel. The same holds true for the
$(\des,\finv)$-generating polynomials and also for the
$(\des,\fmaj)$-generating polynomials for the hyperoctahedral group and the
wreath product groups, confirming further conjectures of Chow and Gessel, and
Chow and Mansour, respectively.
