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It is commonly assumed that pressure and flow in the arteries are purely the result of forward and backward travelling waves. We will show that various observations of arterial behaviour are difficult to explain using this assumption. In particular we will look at what happens to arterial pressure under different conditions: during ectopic beats, in experimental studies of pressure and flow when the aorta is totally occluded at different locations, in a computational study of the input impedances of randomly generated networks of arteries and when pressure and velocity are measured at different distances along the aorta. We show that all of these observations can be explained using wave intensity analysis. We further show that this analysis suggests that it is useful to separate arterial pressure into a reservoir pressure that accounts for the overall compliance of the arterial system and an excess pressure that is determined by local conditions. Evidence has accumulated in the decade since the introduction of the reservoir-wave hypothesis that the reservoir/excess pressure separation can be useful in interpreting the results of vasoactive drugs on cardiovascular performance and that parameters based on the reservoir/excess pressure are significant predictors of cardiovascular events. The important question that remains to be answered is the usefulness of the concept in the interpretation, physiologically and clinically, of the complex behaviour of the cardiovascular system. We conclude that the evidence suggests that it is a worthwhile topic for future research.