This article examines several still disputable aspects of the history of the Crimea and Asian Bosporos (Tmutorokan) in the late tenth and eleventh centuries, and the relations between Byzantium (including imperial administrations in the Crimean Peninsula) and the Tmutorokan Principality, traditionally considered an enclave of Kievan Rus'. Particularly, these aspects include the formation of the Byzantine provincial administration in the Crimea, the themes of Cherson, Sougdaia, and Bosporos, the presence of some areas in Bosporos controlled by the princes of Tmutorokan (the problem of "Crimean Khazaria", the status of the principality in the 1060s-1080s within the Byzantine oecumene). The reason for writing this article was the publication of A. N. Sliadz's monograph, which, in the author's opinion, largely distorts the actual situation in the region because of its very subjective interpretation of numerous old-known facts and the ignorance of new accounts introduced into scholarship in recent years. In this connection, the reviewer have analysed sigillographic monuments uncovering the theme of Sougdaia in the second half of the eleventh century and the evolution of the status of Prince Oleg-Michael of Tmutorokan, from the archon of Matracha, Zichia, and all Khazaria to the archon and doux of Tamatarcha and all Khazaria on the eve of Byzantium's annexation of his possessions (1080s-1090s). Apparently, the latter is documented by the seal of Nikephoros Alanos, vestarches and katepano of Cherson and Khazaria.