  Using fibre bundle theory we construct the universal covering group of U(n),
$\tilde{U}(n)$, and show that $\tilde{U}(n)$ is isomorphic to the semidirect
product $SU(n)\bigcirc {\scriptstyle s}$ R. We give a bijection between the set
of projective representations of U(n) and the set of equivalence classes of
certain unitary representations of $SU(n)\bigcirc {\scriptstyle s}$ R. Applying
Bargmann's theorem, we give explicit expressions for the liftings of projective
representations of U(n) to unitary representations of $SU(n)\bigcirc
{\scriptstyle s}$ R. For completeness, we discuss the topological and group
theoretical relations between U(n), SU(n), U(1) and Z_n.
