  How and when do galaxies form? Studies of the microwave background radiation
reveal that the Universe is spectacularly homogenous at redshift z~1000.
Observations of the local Universe reveal that by z=0 much of the luminous
matter has condensed into mature, gravitationally-bound structures. One of the
primary challenges to astronomers today is to achieve a robust understanding of
this process of galaxy formation and evolution. Locating and studying young
galaxies at large look-back times is an essential aspect of this program. We
review the systematic observational techniques used to identify galaxies at
early cosmic epochs. In the past few years, the study of normal, star-forming
galaxies at z>3 has become possible; indeed, successful methods have been
developed to push the frontier past z=5. We are now directly observing
individual galaxies within a Gyr of the Big Bang. We present a detailed review
of the many search methods used for identifying distant galaxies, consider the
biases inherent in different search strategies, and discuss early results of
these studies. We conclude with goals for future studies at the start of the
21st century.
