The Ramsey Number $R(3,k)$ is that least number $n$ (dependent on $k$) so that {\em any} triangle-free graph on $n$ vertices {\em must} contain an independent set of $k$ vertices. An examination of (appropriately defined) random graphs and random processes plays a key role in finding the asymptotics of $R(3,k)$. Our story takes us from three youngsters, George Szekeres, Esther Klein and Paul Erdos, in the winter of 1931/2 through Greenwood, Gleason, Graver, Yackel, Ajtai, Komlos, Szemeredi, Kim, Lovasz, Winkler, Suen (to name a few!) to very recent work of Bohman.