Superpixels provide a useful intermediate image representation. Existing superpixel methods, however, suffer from at least some of the following drawbacks: 1) topology is handled heuristically; 2) the number of superpixels is either predefined or estimated at a prohibitive cost; 3) lack of adaptiveness. As a remedy, we propose a novel probabilistic model, self-coined Bayesian Adaptive Superpixel Segmentation (BASS), together with an efficient inference. BASS is a Bayesian nonparametric mixture model that also respects topology and favors spatial coherence. The optimizationbased and topology-aware inference is parallelizable and implemented in GPU. Quantitatively, BASS achieves results that are either better than the state-of-the-art or close to it, depending on the performance index and/or dataset. Qualitatively, we argue it achieves the best results; we demonstrate this by not only subjective visual inspection but also objective quantitative performance evaluation of the downstream application of face detection. Our code is available at https://github.com/uzielroy/BASS.