Vector quantization (VQ) is a critical step in representing signals in digital form for computer processing. It has various uses in signal and image compression and in classification. If the signal samples are quantized separately, the operation is called “scalar quantization.” Consequently, if the samples are grouped to form vectors, their quantization is called “vector quantization.” Changing the quantization dimension from one (for scalar) to multiple (for vectors) has many important mathematical and practical implications. VQ produces indices that represent the vector formed by grouping samples. The output index, which is an integer, has little or no physical relation with the vector it is representing, which is formed by grouping real or complex valued samples. The word “quantization” in VQ comes from the fact that similar vectors are grouped together and represented by the same index. Therefore, many distinct vectors on the multidimensional space are quantized to a single vector that is represented by the index. The number of distinct indices defines the number of quantization levels. Assigning indices to a number of vectors has practical applications in compression and classification. This chapter presents the general layout of the VQ operation, introduces VQ design and optimality conditions, and gives examples about compression and classification applications.