Abstract: This paper introduces a framework for post-processing machine learning models so that their predictions satisfy multi-group fairness guarantees. Based on the celebrated notion of multicalibration, we introduce $(s,g,\alpha)-$GMC (Generalized Multi-Dimensional Multicalibration) for multi-dimensional mappings $s$, constraints $g$, and a pre-specified threshold level $\alpha$. We propose associated algorithms to achieve this notion in general settings. This framework is then applied to diverse scenarios encompassing different fairness concerns, including false negative rate control in image segmentation, prediction set conditional uncertainty quantification in hierarchical classification, and de-biased text generation in language models. We conduct numerical studies on several datasets and tasks.