We report the discovery of a substantial population of star–forming galaxies at 3.0 \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } z \lower 2.15 pt \hbox { $ % \buildrel < \over { \sim } $ } 3.5 . These galaxies have been selected using color criteria sensitive to the presence of a Lyman continuum break superposed on an otherwise very blue far-UV continuum , and then confirmed with deep spectroscopy on the W. M. Keck telescope . The surface density of galaxies brighter than { \cal R } = 25 with 3.0 \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } z \lower 2.15 pt \hbox { $ % \buildrel < \over { \sim } $ } 3.5 is 0.4 \pm 0.07 galaxies arcmin ^ { -2 } , approximately 1.3 % of the deep counts at these magnitudes ; this value applies both to “ random ” fields and to fields centered on known QSOs . The corresponding co-moving space density is approximately half that of luminous ( L \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } L ^ { \ast } ) present–day galaxies . Our sample of z > 3 galaxies is large enough that we can begin to detail the spectroscopic characteristics of the population as a whole . The spectra of the z > 3 galaxies are remarkably similar to those of nearby star–forming galaxies , the dominant features being strong low–ionization interstellar absorption lines and high–ionization stellar lines , often with P-Cygni profiles characteristic of Wolf-Rayet and O–star winds . Lyman \alpha emission is generally weak ( < 20 Å rest equivalent width ) and is absent for > 50 % of the galaxies . We assign approximate mass scales to the galaxies using the strengths of the heavily saturated interstellar features and find that , if the line widths are dominated by gravitational motions within the galaxies , the implied velocity dispersions are 180 \leq \sigma \leq 320 km s ^ { -1 } , in the range expected for massive galaxies . The star formation rates , which can be measured directly from the far-UV continua , lie in the range 4 - 25 h _ { 50 } ^ { -2 } M _ { \sun } yr ^ { -1 } ( for q _ { 0 } = 0.5 ) , with 8.5 h _ { 50 } ^ { -2 } M _ { \sun } yr ^ { -1 } being typical . Together with the morphological properties of the z > 3 galaxy population , which we discuss in a companion paper ( Giavalisco et al . 1996 ) , all of these findings strongly suggest that we have identified the high-redshift counterparts of the spheroid component of present–day luminous galaxies . In any case , it is clear that massive galaxy formation was already well underway by z \sim 3.5 .