Wavelets and their associated transforms are highly efficient when approximating and analyzing one-dimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient in sparsely approximating and also in analyzing in the sense of, for instance, detecting their direction. Shearlets are a directional representation system extending the wavelet framework, which overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful implementation and fast associated transforms. In this article, we will introduce a comprehensive carefully documented software package coined ShearLab 3D (www.ShearLab.org) and discuss its algorithmic details. This package provides MATLAB code for a novel faithful algorithmic realization of the 2D and 3D shearlet transform (and their inverses) associated with compactly supported universal shearlet systems incorporating the option of using CUDA. We will present extensive numerical experiments in 2D and 3D concerning denoising, inpainting, and feature extraction, comparing the performance of ShearLab 3D with similar transform-based algorithms such as curvelets, contourlets, or surfacelets. In the spirit of reproducible research, all scripts are accessible on www.ShearLab.org.