The traditional notion of representation for a Lie algebroid or Lie groupoid doesn´t include a natural adjoint representation. It is possible to make sense of the adjoint representation as a "representation up to homotopy," but this requires a choice of connection. In this talk, I will describe models (at both the algebroid and groupoid levels) for adjoint representations that are both canonical and geometrically intuitive. Given a choice of connection, these canonical models naturally decompose into representations up to homotopy.