In this paper we study the theta correspondence for Unitary groups of the same size over local and global fields. This correspondence has been studied in many cases by several authors. We are able to unify and generalise all these known results in terms of two conjectures, one local and the other global. These conjectures are in terms of the parametrisation of irreducible admissible representations of groups over local fields which are formulated by David Vogan refining Langlands parametrization, and which are now called Vogan parameters. In turn, the simple form of the conjecture here, gives support to the importance of Vogan's refinement of Langlands parametrisation.