  If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the
operator $\exp(itA)$. We say that we have perfect state transfer in $X$ from
the vertex $u$ to the vertex $v$ at time $\tau$ if the $uv$-entry of
$|H(\tau)_{u,v}|=1$. This concept has potential applications in quantum
computing. We offer a survey of some of the work on perfect state transfer and
related questions. The emphasis is almost entirely on the mathematics.
