We discuss linear recurrences of Eulerian type of the form P_n(v) = (a(v)n+c(v))P_{n-1}(v) +b(v)(1-v)P_{n-1}'(v) (n > 0), with P_0(v) given, where a(v), b(v) and c(v) are in most cases polynomials of low degree. 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 a(v), b(v) and c(v).We apply our results to more than two hundreds of concrete examples that we collected from the literature and from Sloane's Online Encyclopedia of Integer Sequences. Not only most of the limit results are new, but they are unified in the same framework. The limit laws we worked out include normal, half-normal, Rayleigh, beta, Poisson,negative binomial, Mittag-Leffler, Bernoulli, etc., showing the richness and diversity of such a simple recurrence scheme, as well as the generality and power of the approaches used.