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The measurement of galaxy distances is crucial for some of the basic problems in astronomy and cosmology. In this Chapter I have emphasized the role such measurements play in two of the most important: Hubble constant determination and peculiar velocity analysis. An important distinction between these two efforts, which I have reiterated throughout, is that for peculiar velocities one only needs distances in km s-1, which are independent of an absolute distance scale, whereas for determination of H0 distances in Mpc are required. In practice, this means that peculiar velocity studies may be carried out using distance indicators such as TF or Dn-sigma calibrated only relative to the distant Hubble flow. To obtain H0 the same DIs must be calibrated relative to local galaxies with Cepheid distances. Because the program of Cepheid measurements in local calibrators using HST (Kennicutt et al. 1995) is ongoing, reliable far-field measurements of H0 are still several years away.

I have organized the discussion around the principal distance indicators currently in use. These are:

1. Cepheid variables. The Period-Luminosity relation for these pulsating stars may be calibrated in the Milky Way and in the Magellanic Clouds. However, they are detectable with HST out to ~ 20 Mpc. As such, they will yield accurate absolute distances for ~ 20 local galaxies over the next several years. These local galaxies will in turn provide absolute calibrations for the secondary distances indicators such as TF or SNe Ia that will be used to measure H0 in the ``far field'' (gtapprox 7000 km s-1), where peculiar velocities and depth effects are relatively unimportant.

2. The TF relation. This method has been the workhorse of peculiar velocity studies, for it applies to the ordinary spiral galaxies that best trace the peculiar velocity field. When calibrated using HST Cepheid distances, it promises also to yield a value of H0 accurate to ~ 10%. The TF relation has recently been shown to apply to spiral galaxies at high redshift (Vogt et al. 1996), although evolutionary effects appear to be significant at z appeq 0.5.

3. The Dn-sigma relation. This is a variant of the Fundamental Plane relations for elliptical galaxies. It is comparable to TF in accuracy, and gives similar global results for the large-scale peculiar velocity field (Kolatt & Dekel 1994). Its best chance for absolute calibration comes from a comparison with SBF distances. Like TF, Dn-sigma has recently been applied to relatively high redshift galaxies (Bender et al. 1996), again with evidence of evolutionary changes.

4. Surface Brightness Fluctuations (SBF). This method may be the most accurate DI known for galaxies beyond the range of HST Cepheid measurements, with distance errors as small as 5% under the best conditions and median errors of ~ 8%. Its application is most straightforward for early-type systems, although with care it may be applied to spirals as well. It holds the promise of giving a high-resolution picture of the peculiar velocity field. It will also provide a crucial check of the reliability of TF and Dn-sigma. Its direct application to the H0 problem remains uncertain because of the great technical challenge involved in extending it to distances gtapprox 5000 km s-1.

5. Type Ia Supernovae. SNe are in principle excellent DIs, but suffer from the obvious problem that one cannot, in general, be found in a given galaxy at a given time. In recent years, improved search techniques have vastly increased the number of well-observed SNe Ias, both at relatively low (Hamuy et al. 1995) and high (Perlmutter et al. 1996) redshifts. The results of these studies have included beautiful Hubble diagrams that demonstrate the linearity of the Hubble expansion to z appeq 0.1, with tantalizing hints of curvature that hold the promise of constraining the cosmological parameters Omega0 and Lambda. Sandage and coworkers (Sandage et al. 1996; Saha et al. 1995) have calibrated SNe Ias in galaxies with Cepheid distances to obtain Hubble constant estimates of H0 appeq 57 km s-1 Mpc-1. However, considerable uncertainty attaches to these results at present. The quest for a reliable absolute calibration of SNe Ias continues.

6. The BCG Lm-alpha relation. The pioneering work of Lauer and Postman (Lauer & Postman 1992, 1994; Postman & Lauer 1995) has demonstrated the potential of BCGs in distance scale and peculiar velocity work. The detection of very large-scale bulk streaming using BCGs has caused some to question the global validity of the Lm-alpha relation (e.g., Riess et al. 1995b), but the verdict is not in yet. Ongoing work by Lauer and Postman, in collaboration with Strauss, will greatly clarify the situation.

I conclude by reiterating a point made at the outset of this Chapter. The DIs discussed here are empirical relations whose physical origins are only partially understood at best. There is a class of distance indicators that are based on fairly rigorous physics, and whose absolute calibration may be obtained from first principles. Gravitational lensing of time-variable quasars and the Sunyaev-Zeldovich effect in clusters are perhaps the most noteworthy of these. It is conceivable that these methods will mature in the coming decade and add greatly to what we have learned from the empirical DIs about the distance scale and the peculiar velocity field. However, this additional information will most likely reinforce, %or constrain rather than supplant, the knowledge obtained from the DIs discussed here.

Acknowledgments: I would like to thank David Burstein, Tod Lauer, Marc Postman, Saul Perlmutter, and John Tonry for enlightening discussions about the distance indicator relations in which they are leading experts, and for providing me with data or postscript for several of the figures presented here.

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