We characterize a range of Stochastic Dominance (SD) relations by means of finite systems of convex inequalities. For `SD optimality' of degree 1 to 4 and `SD efficiency' of degree 2 to 5, we obtain exact systems that can be implemented using Linear Programming or Convex Quadratic Programming. For SD optimality of degree five and higher, and SD efficiency of degree six and higher, we obtain necessary conditions. We use separate model variables for the values of the derivatives of all relevant orders at all relevant outcome levels, which allows for preference restrictions beyond the standard sign restrictions. Our systems of inequalities can be interpreted in terms of piecewise polynomial utility functions with a number of pieces that increases with the number of outcomes and the degree of SD. An empirical study analyzes the relevance of higher-order risk preferences for comparing a passive stock market index with actively managed stock portfolios in standard data sets from the empirical asset pricing literature.