How can one formally detect the location of a jump discontinuity in a function? One very successful tool in detecting singularities is the continuous wavelet transform, which I will present. However, the continuous wavelet transform provides no information about the orientation or geometry of the singularity. This is something that we would desire, especially for shape recognition or image restoration/enhancement. Shearlets have proven to be address these weaknesses of wavelets. I will introduce the continuous shearlet transform in R2 and give a brief overview of some recent results in the field. The entire talk is based on a minicourse taught by Demetrio Labate (University of Houston).