The Random Integral Representation Conjecture: a quarter of a century later

In Jurek 1985 and 1988 the random integral representations conjecture was stated. It claims that (some) limit laws can be written as probability distributions of random integrals of the form $\int_{(a,b]}h(t)dY_{\nu}(r(t))$, for some deterministic functions $h$, $r$ and a L\'evy process $Y_{\nu}(t),t\ge 0$. Here we review situations where a such claim holds true. Each theorem is followed by a remark which gives references to other related papers, results as well as some historical comments. Moreover, some open questions are stated.