  Under a non-torsion assumption on Heegner points, results of Kolyvagin
describe the structure of Shafarevich-Tate groups of elliptic curves. In this
paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups
associated with higher weight modular forms over imaginary quadratic fields
satisfying a "Heegner hypothesis". More precisely, we show that the structure
of Shafarevich-Tate groups is controlled by cohomology classes built out of
Nekov\'a\v{r}'s Heegner cycles on Kuga-Sato varieties. As an application of our
main theorem, we improve on a result of Besser giving a bound on the order of
these groups. As a second contribution, we prove a result on the structure of
($p$-primary) Selmer groups of modular forms in the sense of Bloch--Kato.
