Superhydrophobic surfaces, formed by air entrapment within the cavities of hydrophobic solid substrates, offer a promising potential for hydrodynamic drag reduction. In several of the prototypical surface geometries the flows are two-dimensional, governed by Laplace’s equation in the longitudinal problem and the biharmonic equation in the transverse problem. Moreover, low-drag configurations are typically associated with singular limits. Accordingly, the analysis of liquid slippage past superhydrophobic surfaces naturally invites the use of both singular-perturbation methods and conformal-mapping techniques. I will discuss the combined application of these methodologies to several emerging problems in the field.