Survey lecture.

As it is well known, the method of lower and upper solutions
is a very effective tool in order to get existence of solutions
of O.D.E. in given regions of the space, to find invariant
regions, to establish uniqueness and continuous dependence criteria, to
approximate the solution by means of solutions
of linear problems (monotone iterative techniques).
Of course, the classical results have been extended also
to more general settings, such as O.D.E. with delay or O.D.E.
in Banach spaces, but mainly for $C^1$- solutions.
The aim of the lecture is to provide a survey of some rather
recent results regarding comparison theorems for upper and
lower solutions, even under Carathčodory assumptions,
for classical and retarded O.D.E., in finite and infinite
dimensional spaces.