Laue microdiffraction is an X-ray diffraction technique that allows for the non-destructive acquisition of spatial maps of crystallographic orientation and the strain state of (poly)crystalline specimens. To do so, diffraction patterns, consisting of thousands of Laue spots, are collected and analyzed at each location of the spatial maps. Each spot of these so-called Laue patterns has to be accurately characterized with respect to its position, size and shape for subsequent analyses including indexing and strain analysis. In the present paper, several approaches for estimating these descriptors that have been proposed in the literature, such as methods based on image moments or function fitting, are reviewed. However, with the increasing size and quantity of Laue image data measured at synchrotron sources, some datasets become unfeasible in terms of computational requirements. Moreover, for irregular Laue spots resulting, e.g., from overlaps and extended crystal defects, the exact shape and, more importantly, the position are ill-defined. To tackle these shortcomings, a procedure using convolutional neural networks is presented, allowing for a significant acceleration of the characterization of Laue spots, while simultaneously estimating the quality of a Laue spot for further analyses. When tested on unseen Laue spots, this approach led to an acceleration of 77 times using a GPU while maintaining high levels of accuracy.