Since as early as the 19th century, many scholars have devoted themselves to the calculation of sets of orbital elements for particular historical comets. In many cases, these studies have led to proposing orbits that have given satisfactory answers to contemporary observations or later reports about these celestial bodies. As new records or improved translations of existing sources appear, the already calculated orbits can be refined, or even new ones can be achieved. In this paper we focus on historical observations from Eastern and European countries in the late 4th and 5th centuries to suggest new determinations of orbital elements for some of these comets, or, where appropriate, to discuss or improve existing ones. We will also carry out a separate study of comets from the years AD422-423 and AD467, which have been suggested as the parent comets of the Kreutz system of sungrazer comets.