Galaxy distances must be measured by a complicated series of inferences
known as the distance ladder.  We can measure the distances to the
nearest stars by parallax, that is by the apparent motion of the star in
the sky as a result of the Earth's motion round the Sun.  This technique
is limited by the angular resolution that can be obtained.  The
satellite Hipparcos will provide the best measurements, giving the
parallax for around 100,000 stars.  At present parallax can be used
accurately to determine the distances of stars within a few tens of
parsecs from the Sun.  [ 1 parsec = 3.26 lt yrs.]

Statistical methods applied to clusters of stars can be used to extend
the technique further, as can `dynamical parallax' in which the
distances of binary stars can be estimated from their orbital
parameters and luminosities.  In this way, or by other methods, the
distance to the nearest `open clusters' of stars can be estimated;
these can be used to determine a main sequence (unevolved
Hertzsprung-Russell diagram) which can be fitted to other more distant
open clusters, taking the distance ladder out to around 7 kpc.
Distances to `globular clusters', which are much more compact clusters
of older stars, can also have their distances determined in this way
if account is taken of their different chemical composition; fitting
to the H-R diagram of these associations can allow distance estimates
out to 100 kpc.  All of these techniques can be checked against one
another and their consistency verified.

The importance of this determination of distance within our own galaxy
is that it allows us to calibrate the distance indicators that are used
to estimate distances outside it.  The most commonly used primary
distance indicators are two types of periodic variable stars (Cepheids
and RR Lyrae stars) and two types of exploding stars (novae and
supernovae).  Cepheids show a correlation between their period of
variability and their mean luminosity (the colour of the star also plays
a part) so that if the period and magnitude are known the distance can
in principle be calculated.  Cepheids can be observed with ground-based
telescopes out to about 5 Mpc and with the Hubble space telescope to at
least 15 Mpc.  RR Lyrae stars are variables with a well-determined
magnitude; they are too faint to be useful at large distances, but they
allow an independent measurement of the distance to galaxies within 100
kpc, such as the Magellanic Clouds, for comparison with Cepheids.  Novae
show a relationship between luminosity at maximum light and rate of
magnitude decline, though not a very tight one; however, they are
brighter than Cepheids, so this method may allow distance estimates for
more distant objects.  Finally, supernovae allow distance determination
on large scales (since they are so bright), but the method requires some
input from theory on how they should behave as they expand.  The
advantage of using supernovae is that the derived distances are
independent of calibration from galactic measurements; the disadvantage
is that the dependence of the supernova's behaviour on the type of star
that formed it is not completely understood.

The best primary distance indicators (generally Cepheids) can be used
to calibrate mainly empirical secondary distance indicators; these
include the properties of H II regions, planetary nebulae, and
globular clusters in external galaxies and the Tully-Fisher relation
between the width of the 21-cm line of neutral hydrogen and the
absolute magnitude of a spiral galaxy.  These can all be used in
conjunction with type Ia supernovae to push the distance ladder out to
the nearest large cluster of galaxies (Virgo, at around 15--20 Mpc)
and beyond (the next major goal is the Coma cluster at around 5 times
farther away).  Other empirical estimators such as a galaxy
size-luminosity relation or a constant luminosity for brightest
cluster galaxies are of uncertain value.

The goal in all of this is to get out beyond the motions of our local
group of galaxies and determine distances for much more distant
objects which can reasonably be assumed to be moving along with the
expansion of the universe in the Big Bang cosmology.  Since we know
their velocities from their redshifts, this would allow us to
determine Hubble's constant, currently the `holy grail' of
observational cosmology; if this were known we would know the
distances to _all_ distant galaxies directly from their recession
velocity.  Sadly different methods of this determination, using
different steps along the distance ladder, give different results;
this leads to a commonly adopted range for H of between 50 and 100
km/s/Mpc, with rival camps supporting different values.  There are a
number of ongoing attempts to reduce the complexity of the distance
ladder and thus the uncertainty in H.  One has been the recent (and
continuing) use of the Hubble Space Telescope to measure Cepheid
variables directly in the Virgo cluster, thereby eliminating several
steps; this leads to a high (80--100) value of H, although with large
uncertainty (which should hopefully be reduced as more results
arrive).  Other groups are working on eliminating the distance ladder,
with its large uncertainty and empirical assumptions, altogether, and
determining the distances to distant galaxies or clusters directly,
for example using the Sunyaev-Zeldovich effect together with X-ray
data on distant clusters or using the time delays in gravitational
lenses.  The early results tend to support lower values of H, around
50.