Anomaly detection, which aims to identify anomalies deviating from normal patterns, is challenging due to the limited amount of normal data available. Unlike most existing unified methods that rely on carefully designed image feature extractors and memory banks to capture logical relationships between objects, we introduce a text memory bank to enhance the detection of logical anomalies. Specifically, we propose a Three-Memory framework for Unified structural and logical Anomaly Detection (TMUAD). First, we build a class-level text memory bank for logical anomaly detection by the proposed logic-aware text extractor, which can capture rich logical descriptions of objects from input images. Second, we construct an object-level image memory bank that preserves complete object contours by extracting features from segmented objects. Third, we employ visual encoders to extract patch-level image features for constructing a patch-level memory bank for structural anomaly detection. These three complementary memory banks are used to retrieve and compare normal images that are most similar to the query image, compute anomaly scores at multiple levels, and fuse them into a final anomaly score. By unifying structural and logical anomaly detection through collaborative memory banks, TMUAD achieves state-of-the-art performance across seven publicly available datasets involving industrial and medical domains. The model and code are available at https://github.com/SIA-IDE/TMUAD.