  Meyniel's conjecture is one of the deepest open problems on the cop number of
a graph. It states that for a connected graph $G$ of order $n,$ $c(G) =
O(\sqrt{n}).$ While largely ignored for over 20 years, the conjecture is
receiving increasing attention. We survey the origins of and recent
developments towards the solution of the conjecture. We present some new
results on Meyniel extremal families containing graphs of order $n$ satisfying
$c(G) \ge d\sqrt{n},$ where $d$ is a constant.
