  There exist a number of typical and interesting systems or models which
possess three-generator Lie-algebraic structure in atomic physics, quantum
optics, nuclear physics and laser physics. The well-known fact that all simple
3-generator algebras are either isomorphic to the algebra $sl(2,C)$ or to one
of its real forms enables us to treat these time-dependent quantum systems in a
unified way. By making use of the Lewis-Riesenfeld invariant theory and the
invariant-related unitary transformation formulation, the present paper obtains
exact solutions of the time-dependent Schr\"{o}dinger equations governing
various three-generator quantum systems. For some quantum systems whose
time-dependent Hamiltonians have no quasialgebraic structures, we show that the
exact solutions can also be obtained by working in a sub-Hilbert-space
corresponding to a particular eigenvalue of the conserved generator (i.e., the
time-independent invariant that commutes with the time-dependent Hamiltonian).
The topological property of geometric phase factors in time-dependent systems
is briefly discussed.
