  The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field
$F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$.
Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology
groups. Coates-Sujatha observed that the structure of the fine Selmer group of
$E$ over a $p$-adic Lie extension of a number field is intricately related to
some deep questions in classical Iwasawa theory; for example, Iwasawa's
classical $\mu$-invariant vanishing conjecture. In this article, we study the
properties of the $p^\infty$-fine Selmer group of an elliptic curve over
certain $p$-adic Lie extensions of a number field. We also define and discuss
$p^\infty$-fine Selmer group of an elliptic curve over function fields of
characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our
study with a conjecture of Jannsen.
