  This is a survey focusing on the Hasse principle for divisibility of points
in commutative algebraic groups and its relation with the Hasse principle for
divisibility of elements of the Tate-Shavarevich group in the Weil-Ch\^{a}telet
group. The two local-global subjects arose as a generalization of some
classical questions considered respectively by Hasse and Cassels. We describe
the deep connection between the two problems and give an overview of the
long-established results and the ones achieved during the last twenty years,
when the questions were taken up again in a more general setting. In
particular, by connecting various results about the two problems, we describe
how some recent developments in the first of the two local-global questions
imply an answer to Cassel's question, which improves all the results published
before about that problem. This answer is best possible over $\mathbb{Q}$. We
also describe some links with other similar questions, as for examples the
Support Problem and the local-global principle for existence of isogenies of
prime degree in elliptic curves.
