  We investigate the geometric phases and the Bargmann invariants associated
with a multi-level quantum systems. In particular, we show that a full set of
`gauge-invariant' objects for an $n$-level system consists of $n$ geometric
phases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmann
invariants. In the process of establishing this result we develop a canonical
form for U(n) matrices which is useful in its own right. We show that the
recently discovered `off-diagonal' geometric phases [N. Manini and F.
Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in terms
of the basic building blocks developed in this work. This result liberates the
off-diagonal phases from the assumption of adiabaticity used in arriving at
them.
