  The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory
for elliptic curves. The finiteness of the Shafarevich-Tate group of an
elliptic curve $E$ over the field of rational numbers is included in the Birch
and Swinnerton-Dyer conjectures, and is still an open question.
  We present an overview of the Shafarevich-Tate and Selmer groups of an
elliptic curve in the framework of Galois cohomology. Known results on the
finiteness of the Shafarevich-Tate group are mentioned, including results of
Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin.
  We then prove the vanishing of the $\ell$-primary component of the
Shafarevich-Tate group for almost all primes $\ell$, for any elliptic curve $E$
over the rationals without complex multiplication.
