  In this paper, we will study concentration inequalities for Banach
space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly
isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an
Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a
generalization of Pinelis' work on Azuma inequality for martingales with values
in $2$-uniformly smooth space. Secondly, Azuma-type inequality for
self-normalized sums will be presented. Finally, some further inequalities for
Banach space-valued martingales, such as moment inequalities for double indexed
dyadic martingales and the De la Pe\~{n}a-type inequalities for conditionally
symmetric martingales, will also be discussed.
