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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 lowdegrees. We characterize the various limit laws of the coefficients of $P_n(v)$for large $n$ using the method of moments and analytic combinatorial toolsunder varying $\alpha(v), \beta(v)$ and $\gamma(v)$, and apply our results tomore than two hundred of concrete examples when $\beta(v)\ne0$ and more thanthree hundred when $\beta(v)=0$ that we gathered from the literature and fromSloane's OEIS database. The limit laws and the convergence rates we worked outare almost all new and include normal, half-normal, Rayleigh, beta, Poisson,negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprisingrichness and diversity of such a simple framework, as well as the power of theapproaches used.

An asymptotic distribution theory for Eulerian recurrences with applications