L. Carleson introduced the measures which bear his name to solve an interpolation problem for analytic functions (Ann. of Math.,1962), establishing their relationship with the existence of nontangential limits at the boundary. These measures were subsequently understood within the larger context of duality of tent spaces. Carleson measures have played a fundamental role in the theory of elliptic boundary value problems, especially in determining solvability of boundary value problems in the context of non-smooth real or complex coecient operators. The appearance of Carleson measures in this theory is quite natural: solvability in Lp of an elliptic Dirichlet problem is determined by the property of the \weight" or elliptic measure, and weight classes are closely connected to the function space BMO and even have Carleson type characterizations. However, there is an extraordinary variety of ways in which the subtelty of the Carleson measure characterization emerges in elliptic theory. In these lectures, we describe some classical and some modern results in this subject which illustrate this theme.