Boundary discontinuity designs are used to learn about treatment effects along a continuous boundary that splits units into control and treatment groups according to their bivariate score variable. These research designs are also called Multi-Score Regression Discontinuity designs, a leading special case being Geographic Regression Discontinuity designs. We study the statistical properties of commonly used local polynomial treatment effects estimators along the continuous treatment assignment boundary. We consider two distinct approaches: one based explicitly on the bivariate score variable for each unit, and the other based on their univariate distance to the boundary. For each approach, we present pointwise and uniform estimation and inference methods for the treatment effect function over the assignment boundary. Importantly, we show that methods based on univariate distance to the boundary exhibit an irreducible large misspecification bias when the assignment boundary has kinks or other irregularities, making the distance-based approach unsuitable for empirical work in those settings. In contrast, methods based on the bivariate score/location variable do not suffer from this drawback. We illustrate our methods with an empirical application and simulations.