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This book is intended to be the first volume in a series devoted to an in-depth study of medieval European and middle-east comet records. With the aim of covering the entire medieval period, widely understood as corresponding to the 5th to 15th centuries AD, this first volume deals with the 5th, 6th, and 7th centuries. The rest will follow until the period is completed. Comet catalogs are a classic literary genre in the history of astronomy since before the 20th century. In them, the different authors presented reports of observations of different phenomena related to these celestial bodies but always presented a characteristic bias favorable to records from Asia, especially Chinese. This fact is understandable since, in those countries, there was a heritage of systematically writing chronicles of the successive reigns, pointing out astronomical events that, according to their traditions and beliefs, would influence the kingdom or the monarch in some way. This was not the case in Western countries, where we find fewer astronomical observations that are much more dispersed in works by different individual authors who often copy each other or, at least, tend to copy from the most prestigious ones. As a result, to date, there has been no research dedicated to exhaustively studying European literary sources, searching for elements that allow expanding the historical databases on medieval comets, and, at the same time, carrying out astronomical analyses that allow in some cases, the improvement or even the proposal of a set of orbital elements associated with comets.