  We classify up to isomorphism all gradings by an arbitrary group $G$ on the
Lie algebras of zero-trace upper block-triangular matrices over an
algebraically closed field of characteristic $0$. It turns out that the support
of such a grading always generates an abelian subgroup of $G$.
  Assuming that $G$ is abelian, our technique also works to obtain the
classification of $G$-gradings on the upper block-triangular matrices as an
associative algebra, over any algebraically closed field. These gradings were
originally described by A. Valenti and M. Zaicev in 2012 (assuming
characteristic $0$ and $G$ finite abelian) and classified up to isomorphism by
A. Borges et al. in 2018.
  Finally, still assuming that $G$ is abelian, we classify $G$-gradings on the
upper block-triangular matrices as a Jordan algebra, over an algebraically
closed field of characteristic $0$. It turns out that, under these assumptions,
the Jordan case is equivalent to the Lie case.
