Removability of sets for conformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require ``gluing"  of functions to obtain a new  function of the same class. We, therefore, seek geometric conditions on sets that guarantee their removability. In this talk, I will give a survey of old results and discuss some very recent results on the non-removability of the Sierpi\'nski gasket and of Sierpi\'nski carpets. One of the main techniques in my results involves the construction of an abstract metric surface homeomorphic to the Euclidean plane and the embedding of that surface back to the plane with controlled distortion.