  The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph $G$.
  $\bullet$ We present upper and lower bounds for $\mu$ in terms of the
vertex-degree and girth of a transitive graph.
  $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive
cubic graphs (where $\phi$ denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
  $\bullet$ We present strict inequalities for the connective constants
$\mu(G)$ of transitive graphs $G$, as $G$ varies.
  $\bullet$ As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
  $\bullet$ We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
  $\bullet$ A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
  $\bullet$ Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
  $\bullet$ The review closes with a brief account of the "speed" of SAW.
