  This paper begins a series in which we examine the structures of distant
galaxies to directly determine the history of their formation modes. We start
this series by examining the structures of z_F850LP < 27 galaxies in the Hubble
Ultra-Deep field, the deepest high-resolution optical image taken to date. We
investigate a few basic features of galaxy structure using this image. These
include: (1) The agreement of visual eye-ball classifications and
non-parametric quantitative (CAS, Gini/M_20) methods; (2) How distant galaxy
quantitative structures can vary as a function of rest-frame wavelength; and
(3) The evolution of distant galaxy structures up to z~3. One of our major
conclusions is that the majority of galaxies with z_850 < 27 are peculiar in
appearance, and that galaxy assembly is rapidly occurring at these magnitudes,
even up to the present time. We find a general agreement between galaxy
classification by eye and through quantitative methods, as well as a general
agreement between the CAS and the Gini/M_20 parameters. We find that the
Gini/M_20 method appears to find a larger number of galaxy mergers than the CAS
system, but contains a larger contamination from non-mergers. We furthermore
calculate the merger rate of galaxies in the UDF up to z~3, finding an increase
with redshift as well as stellar mass, confirming previous work in the Hubble
Deep Field. We find that massive galaxies with M_{*} > 10^10 M_0 undergo
4.3_+0.8^-0.8 major galaxy mergers at z < 3, with all of this merging occurring
at z > 1.
