Knowing the type (i.e., the biochemical composition) of kidney stones is crucial to prevent relapses with an appropriate treatment. During ureteroscopies, kidney stones are fragmented, extracted from the urinary tract, and their composition is determined using a morpho-constitutional analysis. This procedure is time-consuming (the morpho-constitutional analysis results are only available after several weeks) and tedious (the fragment extraction lasts up to an hour). Identifying the kidney stone type only with the in-vivo endoscopic images would allow for the dusting of the fragments and eneable early treatments, while the morpho-constitutional analysis is ready. Only few contributions dealing with the in vivo identification of kidney stones have been published. This paper discusses and compares five classification methods including deep convolutional neural networks (DCNN)-based approaches and traditional (non DCNN-based) ones. Even if the best method is a DCCN approach with a precision and recall of 98% and 97% over four classes, this contribution shows that an XGBoost classifier exploiting well-chosen feature vectors can closely approach the performances of DCNN classifiers for a medical application with a limited number of annotated data.