  We study linear recurrences of Eulerian type of the form \[
  P_n(v) = (\alpha(v)n+\gamma(v))P_{n-1}(v)
  +\beta(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where
$\alpha(v), \beta(v)$ and $\gamma(v)$ are in most cases polynomials of low
degrees. We characterize the various limit laws of the coefficients of $P_n(v)$
for large $n$ using the method of moments and analytic combinatorial tools
under varying $\alpha(v), \beta(v)$ and $\gamma(v)$, and apply our results to
more than two hundred of concrete examples when $\beta(v)\ne0$ and more than
three hundred when $\beta(v)=0$ that we gathered from the literature and from
Sloane's OEIS database. The limit laws and the convergence rates we worked out
are almost all new and include normal, half-normal, Rayleigh, beta, Poisson,
negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising
richness and diversity of such a simple framework, as well as the power of the
approaches used.
