We introduce region-customizable sound extraction (ReZero), a general and flexible framework for the multi-channel region-wise sound extraction (R-SE) task. R-SE task aims at extracting all active target sounds (e.g., human speech) within a specific, user-defined spatial region, which is different from conventional and existing tasks where a blind separation or a fixed, predefined spatial region are typically assumed. The spatial region can be defined as an angular window, a sphere, a cone, or other geometric patterns. Being a solution to the R-SE task, the proposed ReZero framework includes (1) definitions of different types of spatial regions, (2) methods for region feature extraction and aggregation, and (3) a multi-channel extension of the band-split RNN (BSRNN) model specified for the R-SE task. We design experiments for different microphone array geometries, different types of spatial regions, and comprehensive ablation studies on different system configurations. Experimental results on both simulated and real-recorded data demonstrate the effectiveness of ReZero. Demos are available at https://innerselfm.github.io/rezero/.