With mean separations > 10 h-1 Mpc, clusters of galaxies are ideal objects for sampling efficiently long-wavelength fluctuations over large volumes of the Universe. Furthermore, fluctuations in the cluster distribution are amplified with respect to those in galaxies, i.e. they are biased in much the same way as we were discussing previously some classes of galaxies are: rich clusters form at the peaks of the large-scale density field, and their variance is amplified by a factor that depends on their mass, as it was first shown by Kaiser . In the next section we shall see explicitly this effect at work in recent analyses of the clustering of clusters.
A thorough review of the use of clusters as tracers of large-scale structure has been given recently by Postman , with particular attention paid to optically-selected clusters. For this reason, I will not discuss here the important issue of selecting clusters of galaxies in the optical band, i.e. from the 2D distribution of galaxies on the sky, but I will concentrate on results from X-ray selected cluster samples. Reference to optically-selected samples will be limited to a discussion of the clustering results obtained from them.
Studies of X-ray selected clusters date back to the beginning of X-ray astronomy . However, only in recent years statistical studies became feasible, although still limited to the study of the temperature and luminosity functions, in particular through the data from the Einstein Medium Sensitivity Survey [63, 64]). The ROSAT satellite, launched in 1990, not only produced serendipitous samples of distant clusters to update these studies (see  for a review), but carried out the first all-sky survey ever with an X-ray imaging telescope. This has represented a tremendous input for studies of large-scale structure using clusters of galaxies, as I shall describe in this section.
Figure 7. The distribution on the sky of the 460 X-ray clusters in the REFLEX survey, to fx > 3×10-12 erg s-1 cm-2.
The ROSAT All-Sky Survey (RASS) was performed in the energy band between 0.1 and 2.4 keV with the PSPC, a photon counter with a ~ 20 arcsec resolution on axis, degrading to nearly 2 arcmin at the edges of the 2-degree field of view . The median exposure time in the RASS was of ~ 300 s, which translates in an effective detection limit for clusters of ~ 10-12 erg s-1 cm-2. Considering that in the ROSAT hard band (0.5-2.0 keV), L* 1 x 1044 h-2 erg s-1 , this detection limit translates into a typical depth of dL 900 h-1 Mpc, i.e. z 0.3. The RASS, therefore, provides a unique opportunity to detect clusters of galaxies within a huge volume of the ``local'' Universe.
In fact, follow-up work to construct X-ray cluster samples from the RASS and measure their redshifts started early after completion of the survey (see  for a review). A first example was a survey in the SGP area, that constructed a sample of about 200 clusters with the main aim of measuring the cluster-cluster correlation function . Only recently, however, the all-sky coverage of the RASS was properly exploited. This has been the aim of the ROSAT-ESO Flux Limited X-ray (REFLEX) cluster survey, that uses clusters of galaxies to explore, in the Southern hemisphere, a volume of the Universe comparable to that of the SDSS.
4.1. The REFLEX Survey
The REFLEX cluster survey combines the X-ray data from the RASS and optical follow-up observations using the ESO telescopes, to construct a complete sample of about 700 clusters with measured redshift, to a flux limit fx 1.5 × 10-12 erg s-1 cm-2 in the ROSAT band (0.1-2.4 keV). The survey covers essentially the southern celestial hemisphere ( < 2.5°), at galactic latitudes |bII| > 20°, to avoid regions of high absorption and crowding by stars.
Figure 8. The large-scale distribution of X-ray selected clusters in the REFLEX survey. Only the South Galactic Cap part of the survey is shown here.
During the development of the survey, this project already produced a first bright sample of 135 clusters (the RASS1 bright sample ), limited to the SGP area, that served as a pilot work to fine-tune the strategy. At the time of writing (Spring 1999), the first REFLEX sample of nearly 460 objects with fx > 3× 10-12 erg s-1 cm-2 is on the verge of completion. This first sample, upon which the results on clustering and large-scale structure presented in the next sections are based, has been constructed to be at least 90% complete . Several external checks, as comparisons with independently extracted sets of clusters, support this figure. 95% of the candidates in this sample are confirmed and observed spectroscopically, while all redshifts should be measured by the summer of this year (10) . The distribution on the sky of the REFLEX clusters in this complete sample is shown in Figure 7, while their 3D distribution can be appreciated from the cone diagram of Figure 8. From this latter figure we can see how at this flux limit, the depth of the REFLEX survey is similar to that of the most recent galaxy surveys, as 2dF the and SDSS. At the same time, given the large solid angle of REFLEX, only the SDSS will be able to explore a comparable volume. Of course, clusters provide a coarse-resolution mapping of structures with respect to galaxies, but it is a price one is happy to pay, as in parallel extremely large scales can be explored with a reasonable investment of telescope time.
10 Note added in proof: as of July 1999 only 3 clusters have no measured z. Back.