Conformal dimension of a metric space is the infimum of Hausdorff dimension among all its quasisymmetric images. A space is said to be minimal if its conformal dimension is equal to its Hausdorff dimension. In this talk we will describe some recent results about the conformal dimension of graphs of functions. In particular, we show one dimensional Brownian motion is almost surely minimal for conformal dimension. We also give other examples of sets that are minimal for conformal dimension. These include Bedford-McMullen self-affine carpets with uniform fibers as well as graphs of continuous functions of Hausdorff dimension d, for every d between 1 and 2. This is joint work with Ilia Binder and Wen-Bo Li.