It is a very human tendency to climb a celestial mountain. So it stands for any race, including that of finding individual objects at greater and greater distances, abbreviated usually as at larger redshift, or "bigger z" (where 1 + z = observed / emitted).
As Stern & Spinrad (1999) pointed out in their Table 1, there has been a fairly rapid increase in "zmax" for galaxies; we went from z = 0.20 in 1956 to z =1 by 1982, but then to z = 5.3 in 1998 and z = 5.7 in 1999. The record-breaking progress since 1998 has been due to observations of the strong Ly (from rest 1216 Å) emission line, shifted to the visible and red by the Universal expansion. Over the past year the "LALA" (Large Area Lyman Alpha survey) team (Rhoads et al. 2003) have selected Ly emitters to z = 5.75. They are currently taking images for the z = 6.6 airglow window. Also in 2002 Hu et al. have located a cluster-lensed galaxy at the outstanding redshift of z = 6.56! And as these pages are completed, a Subaru group has found a faint Ly galaxy at z = 6.578.
Modern research on quasars (QSOs, to be more precise), has also progressed; Osmer (1999) reviewed the situation 3 years ago, with QSOs located up to z = 5.0. Since then, the Sloan Digital Sky Survey has been successfully pushed QSO redshifts to and beyond z = 6! The key here is to obtain good red and near IR photometry, in particular looking for objects with very red (I-z) colors. The Sloan results are very current; Fan et al. (2002) found SDSS J103027.1 at z = 6.28, and a preprint on another Sloan QSO at z ~ 6.43 is just available as this section is being written. So the most distant QSO to date still trails the most distant, much fainter normal galaxy by a modest margin!
The distant QSOs are likely buried in a host galaxy which itself is well-hidden in the glare of the Active Galactic Nucleus (AGN). We now assume the presence of the underlying galaxy of stars and gas, in part confirmed indirectly by the normal abundances of the elements inferred from the emission lines in the QSO spectra.