The mathematical ideas underlying capillarity theory were introduced almost two centuries ago. The elegant formal simplicity of the equations is deceptive, and seeming experimental anomalies have led to questioning of their physical adequacy. This talk will describe new studies that show discontinuous dependence of solutions on the boundary data, symmetry breaking, failure of existence under physical conditions, and failure of uniqueness under conditions for which solutions exist. Some of the predictions have been verified in space experiments, and the experimental confirmations offer convincing evidence in support of the physical correctness of the equations as originally formulated. The predictions are suggestive for applications, notably for fluid management under low gravity conditions.