We give new upper and lower bounds on the degree of real multivariate polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of {\em superconstant} depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved {\em constructively}; we give explicit dual solutions to the necessary linear programs.