A treatment of the positive observation problem for positive systems is provided, for the continuous and discrete-time cases. The proposed observers are positive, that is, they ensure that the estimates are nonnegative at any time. Moreover, necessary and sufficient conditions for the existence of such positive observers are formulated in terms of linear programming. Also, we show how positive observers can provide guaranteed bounds on the real states of uncertain positive systems. In addition, some implications are discussed; especially, it is shown that it is not possible to stabilise an unstable system using these positive observers. Finally, the applicability of the proposed interval estimation approach is shown for an illustrative application from pharmacokinetics.