Image compression using memoryless vector quantization (VQ), in which small blocks (vectors) of pixels are independently encoded, has been demonstrated to be an effective technique for achieving bit rates above 0.6 bits per pixel (bpp). To maintain the same quality at lower rates, it is necessary to exploit spatial redundancy over a larger region of pixels than is possible with memoryless VQ. This can be achieved by incorporating memory of previously encoded blocks into the encoding of each successive input block. Finite-state vector quantization (FSVQ) employs a finite number of states, which summarize key information about previously encoded vectors, to select one of a family of codebooks to encode each input vector. In this paper, we review the basic ideas of VQ and extend the finite-state concept to image compression. We introduce a novel for-mulation of the state and state-transition rule that uses a perceptually based edge classifier. We also examine the use of interpolation in conjunction with VQ with finite memory. Coding results are presented for monochrome images in the bit-rate range of 0.24 to 0.32 bpp. The results achieved with finite memory are comparable to those of memoryless VQ at 0.6 bpp and show that there are significant gains to be obtained by enhancing the basic VQ approach with interblock memory.