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The measurement of galaxy distances is one of the most fundamental problems in astronomy. To begin with, we would simply like to know the scale of the cosmos; we do so by determining the distances to galaxies. Beyond this, galaxy distances are the key to measuring the Hubble constant H0, perhaps the most important piece of information for testing the validity of the Big Bang model. Finally, galaxy distances are necessary if we are to study the large-scale peculiar velocity field. Peculiar velocity analysis is among the most promising techniques for confirming the gravitational instability paradigm for the origin of large-scale structure, deducing the relative distributions of luminous and dark matter, and constraining the value of the cosmological density parameter Omega0. In this Chapter, I will describe a number of the methods used for measuring galaxy distances, and discuss their application to the H0 and peculiar velocity problems. When appropriate, I will comment on their relevance to determination of other cosmological parameters as well. The goal of this Chapter is not to present an exhaustive review of galaxy distance measurements, but rather to provide a summary of where matters stand, and an indication of what the next few years may bring.