The Dimension of the Planar Brownian Frontier is 4/3
  In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension of the set of cut points is a.s. 3/4. The present paper is an expository outline of the arguments involved in the proof of these and related results. 
Gregory F. Lawler, Oded Schramm, Wendelin Werner