Proper and strictly proper scoring rules provide a rigorous method for evaluating the accuracy of a probabilistic forecast while encouraging honesty. In this dissertation, we develop new proper and strictly proper scoring rules. We introduce additive and strongly additive scoring rules that can be used to reward a sequence of probabilistic forecasts. We construct new tailored scoring rules and demonstrate a general economic interpretation for all weighted proper scores. We also present a matrix-based construction method for scoring forecasts that can be represented as affine transformations of an underlying distribution.