1. An upper bound for the norm of a system of ordinary differential equations can be obtained by comparison with a related first order differential equation, [4; 8]. This first order equation depends on an upper bound for the norm of the right side of the system. Recently, it has been pointed out [l; 6] that this same upper bound also gives a lower bound for the norm of the solution in terms of another first order equation. Most of the known explicit bounds, as well as criteria for global existence and boundedness, can be obtained from such comparison theorems, together with a detailed analysis of the resulting first order equations. The same approach also yields information on the approach of a solution to a limit. As suggested in [l ], bounds for approximate local solutions can also be obtained in this way. The bounds given in [l ] are sometimes difficult or impossible to calculate explicitly, but it is possible to give slightly weaker bounds, which are more easily calculated. The main conclusion to be drawn from this paper is that the comparison method provides not only a powerful tool for obtaining bounds for solutions, but also a unified approach to many such problems.