  In this paper we investigate the Tate--Shafarevich group Sha^1(k, T) of a
multinorm-one torus $T$ over a global field $k$. We establish a few functorial
maps among cohomology groups and explore their relations. Using these
properties and relations we obtain a few basic structural results for Sha^1(k,
T) and extend a few results of Bayer-Fluckiger--Lee--Parimala [Adv. in Math.,
2019] to some more general multinorm-one tori. We also give a uniform proof of
a result of Demarche--Wei for a criterion of the vanishing of Sha^1(k, T), and
of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the
\'etale $k$-algebra in question is a product of two abelian extensions.
Moreover, we improve the explicit description of Sha^1(k, T) in Lee [J. Pure
Appl. Alg., 2022] by removing an intersection condition.
