Zero-shot anomaly classification and segmentation (AC/AS) aim to detect anomalous samples and regions without any training data, a capability increasingly crucial in industrial inspection and medical imaging. This dissertation aims to investigate the core challenges of zero-shot AC/AS and presents principled solutions rooted in theory and algorithmic design. We first formalize the problem of consistent anomalies, a failure mode in which recurring similar anomalies systematically bias distance-based methods. By analyzing the statistical and geometric behavior of patch representations from pre-trained Vision Transformers, we identify two key phenomena - similarity scaling and neighbor-burnout - that describe how relationships among normal patches change with and without consistent anomalies in settings characterized by highly similar objects. We then introduce CoDeGraph, a graph-based framework for filtering consistent anomalies built on the similarity scaling and neighbor-burnout phenomena. Through multi-stage graph construction, community detection, and structured refinement, CoDeGraph effectively suppresses the influence of consistent anomalies. Next, we extend this framework to 3D medical imaging by proposing a training-free, computationally efficient volumetric tokenization strategy for MRI data. This enables a genuinely zero-shot 3D anomaly detection pipeline and shows that volumetric anomaly segmentation is achievable without any 3D training samples. Finally, we bridge batch-based and text-based zero-shot methods by demonstrating that CoDeGraph-derived pseudo-masks can supervise prompt-driven vision-language models. Together, this dissertation provides theoretical understanding and practical solutions for the zero-shot AC/AS problem.