This paper studies the problem of H-infinity prediction for linear continuous-time systems. By developing a new method of characterizing the innovation process and applying a novel innovation analysis approach in Krein space, a necessary and sufficient condition for the existence of a finite horizon H-infinity predictor is derived. The solution to the H-infinity prediction is given in terms of solutions of Riccati and matrix differential equations. We further extend our study to give a necessary and sufficient condition for the H-infinity filtering of linear continuous-time systems with both instantaneous and delayed measurements. (C) 2004 Elsevier Ltd. All rights reserved.