The defining property of an admissible scoring system is that any individual perceives himself as maximizing his expected score by reporting his true subjective distribution. The use of admissible scoring systems as a measure of probabilistic forecasts is becoming increasingly well-known in those cases where the forecast is a discrete distribution over a finite number of alternatives. Most serious forecasts which are made in the real world seem to be forecasts of quantities rather than choices between a finite number of alternatives. In such cases as this, it seems much more natural to ask the forecaster to specify a continuous probability distribution which represents his expectations rather than trying to re-cast a basically continuous process into a discrete one. To construct an admissible scoring system for a continuous distribution, a collection of possible bets can be postulated on a continuous variable, and an admissible scoring system can be constructed as the net pay-off to a forecaster who takes all bets (and only those bets) which appear favorable on the basis of his reported distribution. Mathematical models for this and alternative systems are presented. (Author/BW)