The authors consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. They present a general procedure which associates to every strongly local vertex operator algebra $V$ a conformal net $mathcal AV$ acting on the Hilbert space completion of $V$ and prove that the isomorphism class of $mathcal AV$ does not depend on the choice of the scalar product on $V$. They show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra $V$, the map $Wmapsto mathcal AW$ gives a one-to-one correspondence between the unitary subalgebras $W$ of $V$ and the covariant subnets of $mathcal AV$.