  The characterization of systems of differential equations admitting a
superposition function allowing us to write the general solution in terms of
any fundamental set of particular solutions is discussed. These systems are
shown to be related with equations on a Lie group and with some connections in
fiber bundles. We develop two methods for dealing with such systems: the
generalized Wei--Norman method and the reduction method, which is very useful
when particular solutions of the original problem are known. The theory is
illustrated with some applications in both classical and quantum mechanics.
