Photometric, number count, and spectrographic evidence is presented to show that most of the blue, starlike objects fainter than mpg = 16"' found in color surveys of high4atitude fields are extragalactic and represent an entirely new class of objects Members of the class called here quasi-stellar galaxies (QSG) resemble the quasi-stellar radio sources (QSS) in many optical properties, but they are radio-quiet. The QSG brighter than `npg = 19"' are 10 times more numerous per square degree than the QSS that are brighter than 9 flux units. The surface density of QSG is about 4 objects per square degree to `npg = 19"'. The evidence is developed in three parts: (1) Photoelectric photometry shows that a fundamental change occurs in the color distribution of high-latitude blue objects at about V = 14."'5. Brighter than this, the objects fall near the luminosity class V line of the U - B, B - V diagram. Fainter than this, 80 per cent of the objects lie in the peculiar region known to be occupied by the quasi-stellar radio sources. (2) The observed integral-count-curve, log N(m), for objects in the Haro-Luyten catalogue undergoes a profound change of slope between `npg = 12"' and `npg = 15"', steepening and reaching a constant slope for mpg fainter than 16"'. This magnitude interval is the same as that in which the color distribution changes, as discussed above. The slope fainter than 16"' is d log N(m)/dm = 0.383. It is shown that this is the expected value from the theory of cosmological number counts for uniformly distributed objects with large redshifts. (3) Spectra of five of the faint blue objects are similar to spectra of quasi-stellar radio sources Intense, sharp emission lines of forbidden [0 iii], [0 ii], and [Ne iii], together with very broad (35 A wide) lines of H , H , H , Ho, and [Ne v] are present in two of the five. Two broad emission lines are present in another at X 3473 and X 4279, identified as C lv (1550) and C iii (1909). The other two objects have featureless spectra with only a blue continuum showing. The redshifts (AX/ 0) for the three objects with lines are 0.0877, 0.1307, and 1.2410. The position of the objects in the redshift-apparentmagnitude diagram shows each of the three to be superluminous. The space density of the quasi-stellar galaxies is estimated to be about 5 X 10-00 QSG/cm3, which is to be compared with the space density of normal galaxies of about 1 X 10-70 galaxies/cm3. The ratio, per unit volume, of QSG to QSS is estimated to be 500, which gives a lifetime of the QSG phase as 5 X 108 years if the lifetime of the radio source is 108 years. The objects would seem to be of major importance in the solution of the cosmological problem. They can be found at great distances because of their high luminosity. QSG at B = 22"' are estimated to have a mean redshift of Ax/x0 5 for a model universe of qo = +1. At these redshifts, we are sampling the universe in depth to 0.63 of the distance to the horizon (for qo = + 1), and are looking back in time more than 0.9 of the way to the "creation event" in an evolutionary model. Study of the [m, z]- and log N(m)curves using the QSG should eventually provide a crucial test of various cosmological models. But even more important, comparative study of the quasi-stellar galaxies and the intimately connected quasistellar radio sources is expected to shed light on the evolutionary processes of the violent events that characterize the two classes.