  Let $f\colon X\to X$ be a continuous function on a compact metric space. We
show that shadowing is equivalent to backwards shadowing and two-sided
shadowing when the map $f$ is onto. Using this we go on to show that, for
expansive surjective maps the properties shadowing, two-sided shadowing,
s-limit shadowing and two-sided s-limit shadowing are equivalent. We show that
$f$ is positively expansive and has shadowing if and only if it has unique
shadowing (i.e.\ each pseudo-orbit is shadowed by a unique point), extending a
result implicit in Walter's proof that positively expansive maps with shadowing
are topologically stable. We use the aforementioned result on two-sided
shadowing to find an equivalent characterisation of shadowing and expansivity
and extend these results to the notion of $n$-expansivity due to Morales.
