The isolated singularity problem for the Yamabe equation has been very well understood since the seminal work of Caffarelli, Gidas and Spruck. In this talk, I will present some results about boundary singularities for elliptic equations with Neumann or Dirichlet conditions. Conformal invariance is a common feature of them. The Neumann problem arises from the recent studies of fractional GJMS operators on the conformal infinity of Poincare-Einstein manifolds. The Dirichlet problem can be viewed as an analogue of Caffarelli-Gidas-Spruck on the boundary, which, however, is open. A partial result will be given. Some new idea here will be used to solve an isoperimetric problem over scalar flat conformal class. This talk partially bases on joint work with L. Caffarelli, T. Jin, O. de Queiroz, Y. Sire and L. Sun.