  Dust-obscured star formation has dominated the cosmic history of star
formation since z = 4. However, the recent finding of significant amount of
dust in galaxies out to z = 8 has opened the new frontier of investigating the
origin of dust also in the earliest phases of galaxy formation, within the
first 1.5 billion years from the Big Bang. This is a key and rapid transition
phase for the evolution of dust, as galaxy evolutionary timescales become
comparable with the formation timescales of dust. Our aim is to provide an
overview of the several findings on dust formation and evolution at z > 4, and
of the theoretical efforts to explain the observational results. We have
organized the review in two parts. In the first part, presented here, we focus
on dust sources, primarily supernovae and asymptotic giant branch stars, and
the subsequent reprocessing of dust in the interstellar medium, through grain
destruction and growth. We also discuss other dust production mechanisms, such
as Red Super Giants, Wolf--Rayet stars, Classical Novae, Type Ia Supernovae,
and dust formation in quasar winds. The focus of this first part is on
theoretical models of dust production sources, although we also discuss the
comparison with observations in the nearby Universe, which are key to put
constraints on individual sources and processes. While the description has a
general applicability at any redshift, we emphasize the relative role of
different sources in the dust build-up in the early Universe. In the second
part, which will be published later on, we will focus on the recent
observational results at z > 4, discussing the theoretical models that have
been proposed to interpret those results, as well as the profound implications
for galaxy formation.
