Let mu be a positive measure supported on a planar domain Omega. We consider the behavior of the balayage measure v := Bal(mu, partial derivative Omega) near a point z(0) is an element of partial derivative Omega at which Omega has an outward- pointing cusp. Assuming that the order and coefficient of tangency of the cusp are d > 0 and a > 0, respectively, and that d mu (z) asymptotic to|z- z(0)|(2b-2)d(2)z as z -> z(0) for some b > 0 (here d(2)z is the Lebesgue measure on C), we obtain the leading order term of v near z(0). This leading term is universal in the sense that it only depends on d, a, and b. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.