A lens space $L$ is the quotient space of an orthogonal free action of a cyclic group of order $p$ on the 3-sphere where $p>2$. In this paper, we determine the isometry groups of these spaces, and show that the spaces can be categorized into six types according to the isomorphism class of the isometry group. Descriptions of these six groups are used to classify up to isomorphism the finite groups which act isometrically and effectively on a lens space. In addition, characterizations are given which describe when a finite group of isometries will preserve a Seifert fibering of the lens space, and when it will respect a genus one Heegaard decomposition.

Keywords: lens space, finite group action, weak equivalence of actions, fiber-preserving actions, orbifold, quotient type